Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Find $\int_{-1}^{1} \frac{\sqrt{4-x^2}}{3+x}dx$ I came across the integral $$\int_{-1}^{1} \frac{\sqrt{4-x^2}}{3+x}dx$$ in a calculus textbook. At this point in the book, only u-substitutions were covered, which brings me to think that there is a clever substitution that one can use to knock off this integral.
I was ab... | HINT:
Using $\displaystyle\int_a^bf(x)\ dx=\int_a^bf(a+b-x)\ dx$
$$I+I=6\int_{-1}^1\dfrac{\sqrt{4-x^2}}{9-x^2}dx$$
Putting $x=2\sin y\implies$
$$2I=6\int_{-\pi/6}^{\pi/6}\dfrac{4\cos^2y}{9-4\sin^2y}dy$$
$$=6\int_{-\pi/6}^{\pi/6}\dfrac{5+4(1-\sin^2y)-5}{9-4\sin^2y}dy$$
$$=6\int_{-\pi/6}^{\pi/6}\left(1-\dfrac5{9-\sin^2y}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1740663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
how to change metric variables The metric on unit sphere is given by:
$$g_{ij}=\begin{pmatrix}1 & 0 \\0 & \sin^2{\theta }\end{pmatrix}.$$
The laplacian beltrami operator in $\theta ,\phi$ is
$$\Delta f=\left(\frac{1}{\sqrt{\vert g\vert}} \partial_{i} \ g^{ij}\sqrt{\vert g\vert}\partial_{j}f\right)$$
$$\Delta f=\left( ... | There are probably more sophisticated (and perhaps "better") ways to do this, but I will stick to very elementary concepts:
The metric expresses how to get arc length. If we embed the sphere into 3-space, the metric $(ds)^2 = (dx)^2 + (dy)^2 + (dz)^2$ matches the given metric on the unit sphere. But if we change $x$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1740986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Characteristic polynomials of matrices related How to show that the characteristic polynomials of matrices A and B are $\lambda^{n-1}(\lambda ^2-\lambda -n)=0$ and $\lambda^{n-1}(\lambda^2+\lambda-n)=0$ respectively by applying elementary row or column operations.
$A=\begin{bmatrix}
1 & 1 & 1 & \cdots & 1 \\
1 & 0 & 0 ... | Use induction on $n$, we only consider the matrix $A$ because solving $B$ is similar. Before starting, I think that the characteristic polynomial of $A$ and $B$ should be of the form $\color{red}{(-1)^{n+1}}\lambda^{n-1}(\lambda^2-\lambda-n)$ and
$\color{red}{(-1)^{n+1}}\lambda^{n-1}(\lambda^2+\lambda-n)$, respectively... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1744698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Finding the power series of a complex function So I have the function
$$\frac{z^2}{(z+i)(z-i)^2}.$$
I want to determine the power series around $z=0$ of this function.
I know that the power series is $\sum_{n=0}^\infty a_n(z-a)^n$, where $a_n=\frac{f^{(n)}(a)}{n!}$. But this gives me coefficients, how can I find a se... | Note that: $$\frac{z^2}{(z+i)(z-i)^2}\equiv \frac{z^2(z+i)}{(z^2+1)^2}$$
This means, that I would need to only find the power series of $\displaystyle\frac1{(z^2+1)^2}$.
We have: $$\frac1{1-x}\equiv\sum_{n\mathop=0}^\infty x^n$$
Taking derivative of both sides: $$\frac{-1}{(1-x)^2}\equiv\sum_{n\mathop=0}^\infty nx^{n-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1746395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Find the solution $(1+2y)dx+(4-x^2)dy=0$
solve $(1+2y)dx+(4-x^2)dy=0$ using separation of variables
$$(1+2y)dx+(4-x^2)dy=0\implies (4-x^2)dy=-(1+2y)dx\implies \frac{dy}{(-1-2y)}=\frac{dx}{(4-x^2)}$$
Integrate the two sides:
$$\begin{align}
-\frac{1}{2}\ln(-2y-1)&=\int \frac{dx}{4-x^2}\\
\\&\text{now,}\\
\int \frac{dx... | Not quite. Note that
$$\int \frac{1}{-1-2y}\,dy=-\frac12 \log|2y+1| \tag 1$$
and
$$\begin{align}
\int \frac{1}{4-x^2}\,dx&=\int \left(\frac{1/4}{2-x}+\frac{1/4}{2+x}\right)\,dx\\\\
&=\frac14 \log\left|\frac{x+2}{x-2}\right|+C \tag 2
\end{align}$$
Then, upon equating $(1)$ and $(2)$, we arrive at
$$-\frac12 \log|2y+1|=... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$m$ order determinant related How to find following $m$ order determinant?
$\begin{vmatrix}
1&1&1&1&1&\cdots&1\\
1&-1&0&0&0&\cdots&0\\
1&0&-1&0&0&\cdots&0\\
1&0&0&-1&0&\cdots&0\\
\vdots&\vdots&\vdots&\vdots&\ddots&\cdots&\vdots\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\
1&0&0&0&0&\vdots&-1\\
\end{vmatrix}_m$ ... | Recall that the determinant does not change if we add one row to another.
By adding all the rows to the first row, we obtain a diagonal matrix whose determinant is easy to evaluate
$\begin{align} \begin{vmatrix}
1&1&1&1&1&\cdots&1\\
1&-1&0&0&0&\cdots&0\\
1&0&-1&0&0&\cdots&0\\
1&0&0&-1&0&\cdots&0\\
\vdots&\vdots&\vdots&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1749071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can't seem to solve a radical equation? Question is : $\sqrt{x+19} + \sqrt{x-2} = 7$ So there is this equation that I've been trying to solve but keep having trouble with.
The unit is about solving Radical equations and the question says
Solve:
$$\sqrt{x+19} + \sqrt{x-2} = 7$$
I don't want the answer blurted, I want... | $\sqrt{x+19} + \sqrt{x-2} = 7$
Squaring both sides, we have
$x+19+2\sqrt{x+19}\sqrt{x-2}+x-2=49$
Collecting terms, we have
$2x+17+2\sqrt{x^2+17x-38}=49$
$\sqrt{x^2+17x-38}=\dfrac{32-2x}{2}$
Squaring again
$x^2+17x-38=\dfrac{1024-128+4x^2}{4}$
$x^2+17x-38=256-32x+x^2$
$49x=294$
$\therefore x=\dfrac{294}{49}=6$
We can... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1750192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
How to square both the sides of an equation? Question: $x^2 \sqrt{(x + 3)} = (x + 3)^{3/2}$
My solution: $x^4 (x + 3) = (x + 3)^3$
$=> (x + 3)^2 = x^4$
$=> (x + 3) = x^2$
$=> x^2 -x - 3 = 0$
$=> x = (1 \pm \sqrt{1 + 12})/2$
I understand that you can't really square on both the sides like I did in the first step, howe... | We have
$$x^2\sqrt{x+3} = \sqrt{x+3}\cdot \vert x+3 \vert \implies \sqrt{x+3}\left(x^2-\vert x+3 \vert\right) = 0$$
In the previous statement, we made use of the fact that
$$(x+3)^{3/2} = \sqrt{x+3}\cdot \vert x+3 \vert$$
Hence, we have either
*
*$\sqrt{x+3} = 0 \implies x = -3$
*$x^2-x-3 = 0$ and $x+3 > 0$. This i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1751410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Integrating $\int \frac{\sqrt{x^2-x+1}}{x^2}dx$ Evaluate $$I=\int\frac{\sqrt{x^2-x+1}}{x^2}dx$$ I first Rationalized the numerator and got as
$$I=\int\frac{(x^2-x+1)dx}{x^2\sqrt{x^2-x+1}}$$ and splitting we get
$$I=\int\frac{dx}{\sqrt{x^2-x+1}}+\int\frac{\frac{1}{x^2}-\frac{1}{x}}{\sqrt{x^2-x+1}}dx$$ i.e.,
$$I=\int\fr... | Using Integration by Parts, We get
Let $$I =\int \sqrt{x^2-x+1}\cdot \frac{1}{x^2}dx = -\frac{\sqrt{x^2-x+1}}{x}+\int\frac{2x-1}{2x\sqrt{x^2-x+1}}dx$$
]
So we get $$I = -\frac{\sqrt{x^2-x+1}}{x}+\int\frac{1}{\sqrt{x^2-x+1}}dx-\frac{1}{2}\int\frac{1}{x\sqrt{x^2-x+1}}dx$$
Now for last Integral, Let $$J = \int \frac{1}{x\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1753855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Projectile motion: Proving:$ x^2 + 4 \left(y-\frac{v^2}{4g} \right)^2 = \frac{v^2}{4g^2} $
Question: Projectiles are fired with initial speed $v$ and variable launch angle $0< \alpha < \pi$.
Choose a coordinate system with the firing position at the origin. For each value of $\alpha$ the trajectory will follow a para... | We have:
$$4\left(y-\frac{v^2}{4g}\right)^2=4\left(\frac{v^2\sin^2(\alpha)}{2g}-\frac{v^2}{4g}\right)^2=\frac{v^4\sin^4(\alpha)}{g^2}-\frac{v^4\sin^2(\alpha)}{g^2}+\frac{v^4}{4g^2}$$
$$=\left(\frac{v^4\sin^2\alpha}{g^2}\right)(\sin^2\alpha-1)+\frac{v^4}{4g^2}$$
$$=-\left(\frac{v^4\sin^2\alpha}{g^2}\right)(1-\sin^2\alph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1754894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How do you prove this without using induction? How do you prove this without using induction
$$\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n-1}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots+\frac{1}{2n-1}$$
| Let $A(n)=\frac 11 + \frac 12 + \frac 13 + \dots + \frac 1n$. We have
$$
\begin{align}
A(2n)&=\frac 11 + \frac 12 + \frac 13 + \dots + \frac 1n + \dots + \frac 1{2n} \\
&=\left(\frac 11 + \frac 13 + \frac 15 + \dots + \frac 1{2n-1}\right)+\left(\frac 12 + \frac 14 + \frac 16 + \dots + \frac 1{2n}\right) \\
&=\left(\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1755083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Symmetric system of equations problem Solve the following simultaneous eqations on the set of real numbers: $$a^2+b^3=a+1$$ $$b^2+a^3=b+1$$
I have found two trivial solutions: $$a=b=1$$ $$a=b=-1$$
but I can't prove that there are no others.
| Subtracting the former from the latter gives
$$b^2-a^2+a^3-b^3=b-a,$$
i.e.
$$(b-a)(b+a)+(a-b)(a^2+ab+b^2)=b-a$$
$$(b-a)(b+a-a^2-ab-b^2-1)=0$$
Case 1 :
If $b=a$, then
$$a^2(a+1)=a+1\iff (a-1)(a+1)^2=0\iff a=\pm 1$$
Case 2 :
If $b+a-a^2-ab-b^2-1=0$, then
$$b^2+(-1+a)b-a+a^2+1=0$$
Since $b$ is real, we have to have
$$(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1756278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that $\frac{z}{1-z} = \sum_{j=0}^∞ \frac{2^j}{1 + z^{-2^j}}$ when $z ∈ \mathbb{D}$ The question
Knowing that with $z ∈ \mathbb{D}$:
$$ \prod_{k=0}^∞(1 + z^{2^k}) = \frac{1}{1-z} $$
prove that with $z ∈ \mathbb{D}$:
$$ \sum_{j = 0}^∞ \frac{2^j}{1 + z^{-2^j}} = \frac{z}{1-z} $$
What I've tried
$$ \sum_{j = 0}^∞ \fra... | Taking the "logarithmic derivative" $f'/f = (\log f)'$ on both
sides of the equation
$$
\prod_{k=0}^\infty (1 + z^{2^k}) = \frac{1}{1-z}
$$
gives
$$
\sum_{k=0}^\infty \frac{2^k z^{2^k-1}}{1 + z^{2^k}} = \frac{1}{1-z}
$$
which is the desired result.
See $\frac{f'(z)}{f(z)}= \sum_{n=1}^{+ \infty}\frac{f'_{n}(z)}{f_{n}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1756403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Generators for a matrix group Let's denote $\Gamma_0(4)$ the subgroup of $\mathrm{SL}_2(\mathbb Z)$:
$$\Gamma_0(4):=\left\{\begin{pmatrix} a &b\\ 4c&d \end{pmatrix}\in \mathrm{SL}_2(\mathbb Z)\right\}.$$
We also define $A$ and $B$ in $\Gamma_0(4)$ as follow:
$$\gamma_1:=\begin{pmatrix} 1 &1\\ 0&1 \end{pmatrix},$$
$$\ga... | Since I managed to find the answer, I am going to write it here in case it is useful for someone.
Let's defined the group $G=<\gamma_1,\gamma_2>$ generated by:
$$ \displaystyle \gamma_1=\begin{pmatrix} 1 & 0\\ 4 & 1 \end{pmatrix} \text{ and } \displaystyle \gamma_2=\begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix}.$$
Fi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1756619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
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Suppose $\triangle ABC$ is an equilateral triangle inscribed in the unit circle C(0,1). Suppose $\triangle ABC$ is an equilateral triangle inscribed in the unit circle C(0,1). Find the maximum value of $$\overline{PA}\cdot\overline{PB}\cdot\overline{PC}$$
where $P$ is a variable point in $\bar{D}(0,2).$
I am trying to ... | Supposing
$$\triangle ABC =
\left((0,1) ,\,
\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2} \right) ,\,
\left(+\frac{\sqrt{3}}{2},-\frac{1}{2} \right) \right)$$
and $P =(x,y) \in D(0,2) = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 \leq 2\}$ and $$\Pi^2 = \overline{PA}\cdot\overline{PB}\cdot\overline{PC},$$
we write
\begin{align}
\P... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1758240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Tridiagonal matrix inner product inequality I want to show that there is a $c>0$ such that
$$
\left<Lx,x\right>\ge c\|x\|^2,
$$
for alle $x\in \ell(\mathbb{Z})$ where
$$
L=
\begin{pmatrix}
\ddots & \ddots & & & \\
\ddots & 17 & -4 & 0 & \\
\ddots & -4 & 17 & -4 & \ddots \\
& 0 & -4 & 17 & \ddots \\
... | If $V$ is the bilateral shift, we have $$L=17 I - 4(V+V^*).$$
From $\|V\|=1$, we get that $V+V^*$ is a selfadjoint with $\|V+V^*\|\leq2$. Then, for a unit vector $x$,
$$
\langle Lx,x\rangle=17-4\langle(V+V^*)x,x\rangle\geq17-4\|V+V^*\|\geq 17-8=9.
$$
In other words, you can take $c=9$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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arithmetic mean of smallest numbers of all subsets of r elements formed out of (1,2,..n) Consider all subsets of r elements of the set $\{1,2,3,......,n\}$ where $1 \leq r \leq n$.
Each of these subsets has a smallest member. Let $F(n,r)$ denote the arithmetic mean of these smallest numbers then
$$
F(n,r)=\frac{n+1}{r+... | The number of ways that $k$ is the smallest of $r$ numbers from $1\dots n$ is
$$
\binom{n-k}{r-1}\tag{1}
$$
As one would expect, the total number of ways to arrange the $r$ numbers from $1\dots n$ is
$$
\sum_{k=1}^{n-r+1}\binom{n-k}{r-1}=\binom{n}{r}\tag{2}
$$
Thus, the expected smallest of $r$ numbers from $1\dots n$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove: $\arcsin\left(\frac 35\right) - \arccos\left(\frac {12}{13}\right) = \arcsin\left(\frac {16}{65}\right)$ This is not a homework question, its from sl loney I'm just practicing.
To prove :
$$\arcsin\left(\frac 35\right) - \arccos\left(\frac {12}{13}\right) = \arcsin\left(\frac {16}{65}\right)$$
So I changed all t... | If $0<x<1$, then both $\arcsin x$ and $\arccos x$ are in $(0,\pi/2)$. I'll assume $0<x<1$ for the rest of the discussion.
If $\alpha=\arcsin x$, then $\sin\alpha=x$ and $\cos\alpha=\sqrt{1-x^2}$; therefore
$$
\tan\alpha=\frac{x}{\sqrt{1-x^2}}
$$
and
$$
\arcsin x=\alpha=\arctan\frac{x}{\sqrt{1-x^2}}
$$
Similarly, if $\b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1760940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Prove: $\frac{a+c}{b+d}$ lies between $\frac{a}{b}$ and $\frac{c}{d}$ (for positive $a$, $b$, $c$, $d$) I am looking for proof that, if you take any two different fractions and add the numerators together then the denominators together, the answer will always be a fraction that lies between the two original fractions.
... | Assume that $b,d >0$. Note that $$\frac{a+c}{b+d} = \frac{b}{b+d}\frac{a}{b} +\frac{d}{b+d}\frac{c}{d}.$$ Remark that $0<\frac{b}{b+d}<1$ and the same for $\frac{d}{b+d}$. Hence you have written $\frac{a+b}{c+d}$ as a convex combination of $\frac{a}{b}$ and $\frac{c}{d}$ so you get $$\frac{a}{b} < \frac{a+c}{b+d} < \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1761557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 3
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Evaluate $\int_{C[0,3]} \frac{\exp(z)}{(z+2)^2\sin(z)} dz$ Using Residue Theorem $\displaystyle \int_{C[0,3]} \frac{\exp(z)}{(z+2)^2\sin(z)} \, dz$ using Residue Theorem.
I have found singularities within $C[0,3]$, which are $-2$ and $0$. For $z=-2$, it is a pole with degree $2$. However, I do not know what kind of sin... | To answer the specific issue raised in the OP, the pole at $z=0$ is a simple pole since $\sin(z)=z(1+O(z^2))$ and therefore $\frac{1}{\sin(z)}=\frac{1}{z(1+O(z^2))}=\frac{1}{z}+O(1)$
Now, as stated in the comments, there are a number of ways forward to determining the residue at $z=-2$.
METHODOLGY $1$: Apply Standar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1764849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Cosine Inequality Show that given three angles $A,B,C\ge0$ with $A+B+C=2\pi$ and any positive numbers $a,b,c$ we have $$bc\cos A + ca \cos B + ab \cos C \ge -\frac {a^2+b^2+c^2}{2}$$
This problem was given in the course notes for a complex analysis course, so I anticipate using $$bc\cos A + ca \cos B + ab \cos C=\math... | Take three vectors $\mathbf a, \mathbf b,$ and $\mathbf c$. We have
$$(\mathbf a + \mathbf b + \mathbf c) \cdot (\mathbf a + \mathbf b + \mathbf c) \ge 0,$$
so
$$\lVert \mathbf a\rVert^2 + \lVert \mathbf b\rVert^2 + \lVert \mathbf c\rVert^2 + 2(\mathbf a \cdot \mathbf b) + 2(\mathbf a \cdot \mathbf c) + 2(\mathbf b \cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1766099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
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Find the value of $\frac{1}{a^3}+\frac{1}{b^3}$ Let be a, b solutions for the equation $x^2-2\sqrt{\sqrt{2}+1} (x)+\sqrt{\sqrt{2}+1}=0$
Find the value of $\frac{1}{a^3}+\frac{1}{b^3}$
Using Vieta's formula $a+b=2c$
, $ab=c$ (where $c=\sqrt{\sqrt{2}+1}$) and solving I find that the answer is
$\frac{1}{a^3}+\frac{1}{b^... | Let $ c = \sqrt{\sqrt{2} + 1} $. Then your given equation is written as :
$ x^2 - 2cx + c = 0 $
Solving it gives you : $ Δ = 4c^2 - 4c > 0 $
$ x_{1,2} = \frac {- 2c \pm \sqrt{Δ}}{2} $
You can then proceed with the calculations by substituting $c$ back in. Just be careful with the square roots and your calculations :)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1769099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Isomorphism: $F = \mathbb{Z}_{5}(\alpha)$, $\alpha^2 +2 =0$, and $F'= \mathbb{Z}_{5}(\beta)$, $\beta ^2 + \beta + 1 = 0$. Let $F = \mathbb{Z}_{5}(\alpha)$, where $\alpha^2 +2 =0$, and let $F'= \mathbb{Z}_{5}(\beta)$, where $\beta ^2 + \beta + 1 = 0$.
Exhibit the isomorphism between $F$ and $F'$.
Honestly, I don't know... | Note that $\alpha^2 = 3$, thus $\alpha^4 = 3^2 = 4$, and so $\alpha^8 = 4^2 = 1$. So the order of $\alpha$ is 8.
Now $(\alpha + 1)^2 = \alpha^2 + 2\alpha + 1 = 3 + 2\alpha + 1 = 2\alpha + 4$, and consequently:
$(\alpha + 1)^3 = (\alpha + 1)(2\alpha + 4) = 2\alpha^2 + 4\alpha + 2\alpha + 4 = 2(3) + 4 + \alpha = \alpha$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1770621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Find real roots of the equation
Find all real solutions to
$$\dfrac{\sqrt{x+1}}{2+\sqrt{2-x}} - \dfrac{\sqrt{x^2-x+2}}{2+\sqrt{-x^2+x+1}} = x^3-x^2-x+1$$
This question is very similar to one of my previous problem, except I cannot find a monotonic function as has been done in the solution to that problem.
Any help wi... | This really is similar in some way to your previous problem.
$$\dfrac{\sqrt{x+1}}{2+\sqrt{2-x}} - \dfrac{\sqrt{x^2-x+2}}{2+\sqrt{-x^2+x+1}} = x^3-x^2-x+1$$
Firstly, for all the terms to be valid, we do domain checking. Solving $x+1\ge0$, $2-x\ge0$, $x^2-x+2\ge0$ and $-x^2+x+1\ge0$, we arrive at $x\in[-0.618, 1.618]$ (a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1770880",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Prove this inequality with $a+b+c=3$ Let $a,b,c>0$,and $a+b+c=3$,show that
$$\dfrac{a}{2b^3+c}+\dfrac{b}{2c^3+a}+\dfrac{c}{2a^3+b}\ge 1$$
such Use Cauchy-Schwarz inequality we have
$$\left(\dfrac{a}{2b^3+c}+\dfrac{b}{2c^3+a}+\dfrac{c}{2a^3+b}\right)\left(a(2b^3+c)+b(2c^3+a)+c(2a^3+b)\right)\ge (a+b+c)^2=9$$
Therefore,i... | By C-S $\sum\limits_{cyc}\frac{a}{2b^3+c}=\sum\limits_{cyc}\frac{a^2(a+c)^2}{a(a+c)^2(2b^3+c)}\geq\frac{\left(\sum\limits_{cyc}(a^2+ab)\right)^2}{\sum\limits_{cyc}a(a+c)^2(2b^3+c)}$.
Hence, it remains to prove that $(a+b+c)^2\left(\sum\limits_{cyc}(a^2+ab)\right)^2\geq9\sum\limits_{cyc}a(a+c)^2(2b^3+c)$, which is
$\sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1772228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Maximum value of the sum of absolute values of cubic polynomial coefficients $a,b,c,d$
If $p(x) = ax^3+bx^2+cx+d$ and $|p(x)|\leq 1\forall |x|\leq 1$, what is the $\max$ value of $|a|+|b|+|c|+|d|$?
My try:
*
*Put $x=0$, we get $p(0)=d$,
*Similarly put $x=1$, we get $p(1)=a+b+c+d$,
*similarly put $x=-1$, we get ... | Let $A=\max(|a|,|c|),C=\min(|a|,|c|),B=\max(|b|,|d|),D=\min(|b|,|d|).$ Then $$|A|+|B|+|C|+|D|=|a|+|b|+|c|+|d|.$$
For $|x|\le1$, $|Ax^2-C|\le|ax^2+c|$ and $|Bx^2-D|\le|bx^2+d|$. Then
$$|(A+B)x^3-(C+D)x|\le |Ax^3-Cx|+|Bx^2-D|\le |ax^3+cx|+|bx^2+d|\le |p(x)|\text{ or } |p(-x)|.$$
Therefore we only need consider $p(x)=ax^3... | {
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"url": "https://math.stackexchange.com/questions/1773846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
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Finding the Exponential of a Matrix that is not Diagonalizable Consider the $3 \times 3$ matrix
$$A =
\begin{pmatrix}
1 & 1 & 2 \\
0 & 1 & -4 \\
0 & 0 & 1
\end{pmatrix}.$$
I am trying to find $e^{At}$.
The only tool I have to find the exponential of a matrix is to diagonalize it. $A$'s eigenvalue is 1. Therefore, ... | (This question was edited a lot, I'm referring to this revision.)
Yes, this is correct. Note however that:
*
*You've used $e^{(M+N)t}=e^{Mt}e^{Nt}$. Note that this is only valid if $M$ and $N$ commute (that is, $MN=NM$). In this case it's ok because $M$ is scalar and commutes with everything, but you should mention ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1775469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 7,
"answer_id": 2
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integration using change of variables
find $$\iint_{R}x^2-xy+y^2 dA$$ where $R: x^2-xy^+y^2=2$ using $x=\sqrt{2}u-\sqrt{\frac{2}{3}}v$ and $y=\sqrt{2}u+\sqrt{\frac{2}{3}}v$
To calculate the jacobian I take $$\begin{vmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v}\\
\frac{\partial y}{\partial u}... | Let be $$I=\iint_{R}(x^2-xy+y^2)\, \mathrm dA$$ where $R: x^2-xy+y^2=2$.
Using the change of variables $x=\sqrt{2}u-\sqrt{\frac{2}{3}}v$ and $y=\sqrt{2}u+\sqrt{\frac{2}{3}}v$ the domain of integration $R$ becomes $S:u^2+v^2=1$ and the integrand function $x^2-xy+y^2$ becomes $2(u^2+v^2)$. The Jacobian determinant is
$$\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1779468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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If $a$ is not divisible by $7$, then $a^3 - 1$ or $a^3 + 1$ is divisible by $7$ Determine is, in general, true or false. Recall that a universal statement is true if it is true for all possible cases while it is false if there is even one counterexample. Be prepared to prove that your answer is correct by supplying a p... | It's correct. An alternative can be:
You could realize that
$$x^3 \equiv 0, 1, -1 \pmod 7$$
So, if $7 \nmid x$ then $x^3 \equiv 1, -1 \pmod 7$.
By the definition of congruences we have, either $x^3 + 1 = 7k$ or $x^3 - 1 = 7k$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1779583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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If $3^x +3^y +3^z=9^{13}$.Find value of $x+y+z$ Problem: If $3^x +3^y +3^z=9^{13}$.Find value of $x+y+z$.
Solution: $3^x +3^y +3^z=9^{13}$
$3^x +3^y +3^z=3^{26}$
I am unable to continue from here.
Any assistance is appreciated.
Edited
$9^{13} =3^{26}$
$=3^{25} (3)$
$=3^{25} (1+1+1)$
$=3^{25} + 3^{25} + 3^{25}$
So $x+y... | If $x=y=z$ then clearly $x=y=z=25$ works.
Otherwise, this is a ternary number with sum of digits $3$. But the unique representation of this number in ternary is
$$1\underbrace{0 \ldots 0}_{26 \text{ times}}$$
which has a sum of digits as $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1779912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Find all integral solutions of the equation $x^n+y^n+z^n=2016$
Find all integral solutions of equation
$$x^n+y^n+z^n=2016,$$
where $x,y,z,n -$ integers and $n\ge 2$
My work so far:
1) $n=2$ $$x^2+y^2+z^2=2016$$
I used wolframalpha n=2 and I received the answer to the problem (Number of integer solutions: 144)
2) ... | Here an incomplete answer about the possibilities for $2016$. It is clear that the problem would become more difficult for higher numbers.
$$\boxed {n=2}$$
It is known enough that an integer $n$ is a sum of three squares if and only if $n$ is not of the form $4^a(8b+7)$ so because $2016=4^2(8\cdot15+6)$ we can ensure t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1780881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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What is the probability of selecting five of the winning balls and one of the supplementary balls? So I'm just doing a bit of probability questions and wanted to make sure I got it right.
I have $50$ balls numbered $1-50$, and we pick $6$ winning balls and $2$ supplementary without replacement.
So the chance to get t... |
So the chance to get one of the $6$ winning balls would simply be: $\frac{6}{50}*\frac{5}{49}*\frac{4}{48}*\frac{3}{47}*\frac{2}{46}*\frac{1}{45} = \frac{1}{15890700}$
No, that is the chance to pick all of the six winning balls when you pick six numbers. Exactly one winning ball among six picks would be:
$$\frac{6}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1781692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Cyclic Inequality in 3 variables How can I prove the following inequality
$$\frac{2a}{1+b^2}+\frac{2b}{1+c^2}+\frac{2c}{1+a^2}\geq 3, \forall\ a,b,c>0, a+b+c=3.$$
I tried Cauchy inequality, AM-GM, but I don't get anything good...
| By AM-GM $\sum\limits_{cyc}\frac{2a}{1+b^2}=\sum\limits_{cyc}\left(\frac{2a}{1+b^2}-2a\right)+6=\sum\limits_{cyc}\frac{-2ab^2}{1+b^2}+6\geq\sum\limits_{cyc}\frac{-2ab^2}{2b}+6=$
$=6-(ab+ac+bc)\geq6-3=3$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1782610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Evaluate $\int\frac{\sqrt{x^2+2x-3}}{x+1}d\,x$ by trig substitution I am preparing for an exam and found this integral in a previous test. Did I do it correctly?
My attempt.
$$
\int\frac{\sqrt{x^2+2x-3}}{x+1}\,dx
$$
Complete the square of $x^2+2x-3$; I changed the integral to
$$
\int\frac{\sqrt{(x-1)^2-4}}{x+1}\,dx
$$
... |
I changed the integral to $\int\frac{\sqrt{(x-1)^2-4}}{x+1}dx$ then $u=x+1$
It should be
$$\int\frac{\sqrt{(x\color{red}{+}1)^2-4}}{x+1}dx$$
I simplified to $2\int tan^2\theta d\theta$ = $2\int\sec^2\theta-1d\theta$
$=2[tan\theta-\theta]+C$
back substitute $\theta=tan^{-1}\frac{\sqrt{u^2-4}}{2}$and $tan\theta=\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1783684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Evaluation of $\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7}$
Evaluation of $$\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7} = $$
$\bf{My\; Try::}$ I have solved Using Direct formula::
$$\sin \frac{\pi}{n}\cdot \sin \frac{2\pi}{n}\cdot......\sin \frac{(n-1)\pi}{n} = \frac{... | $$
\begin{align}
\prod_{k=1}^3\sin\left(\frac{k\pi}7\right)^2
&=\prod_{k=1}^6\sin\left(\frac{k\pi}7\right)\tag{1}\\
&=-\frac1{64}\prod_{k=1}^6\left(e^{ik\pi/7}-e^{-ik\pi/7}\right)\tag{2}\\
&=\frac1{64}\prod_{k=1}^6\left(1-e^{-i2k\pi/7}\right)\tag{3}\\
&=\frac1{64}\lim_{z\to1}\prod_{k=1}^6\left(z-e^{-i2k\pi/7}\right)\ta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1784712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
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Exercise: Evaluate a polynomial function such as $P(x)=2x^3-3x^2+7x-2$ at a surd such as $x=1+2\sqrt{3}$. Exercise: Given polynomial function $P(x)=2x^3-3x^2+7x-2$ evaluate $P(x)$ at the surd $x=1+2\sqrt{3}$.
| Solution: Divide $P(x)$ by the divisor
\begin{equation}
D(x)=[x-(1+2\sqrt{3})]\cdot[x-(1-2\sqrt{3})]=x^2-2x-11
\end{equation}
Then
\begin{equation}
P(x)=(2x+1)(x^2-2x-11)+31x+9
\end{equation}
Therefore
\begin{equation}
P(1+2\sqrt{3})=0+31(1+2\sqrt{3})+9=40+62\sqrt{3}
\end{equation}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1786511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to calculate $\lim_{x\to 0}\left(\frac{1}{x^2} - \frac{1}{\sin^2 x}\right)^{-1}$? $$f (x) = \frac{1}{x^2} - \frac{1}{\sin^2 x}$$
Find limit of $\dfrac1{f(x)}$ as $x\to0$.
| $$\frac{1}{\sin^2 x}=\frac{1}{(x-x^3/6+O(x^5))^2}=\frac{1}{x^2}\cdot\frac{1}{(1-x^2/6+O(x^4))^2}\\=\frac{1}{x^2}\cdot\frac{1}{(1-x^2/3+O(x^4))}=\frac{1}{x^2}(1+x^2/3+O(x^4))$$
Hence
$$f(x)=\frac{1}{x^2}-\frac{1}{\sin^2 x}=-\frac13+O(x^2)$$
And
$$\frac{1}{f(x)}\underset{x\to 0}{\longrightarrow} -3$$
Without using Taylo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1786681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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How does one parameterize $x^2 + xy + y^2 = \frac{1}{2}$?
Parameterize the curve $C$ that intersects the surface
$x^2+y^2+z^2=1$ and the plane $x+y+z=0$.
I have this replacing equations:
$$ x^2+y^2+(-x-y)^2=1$$
and clearing have the following:
$$ x^2+xy+y^2=1/2$$
which it is the equation of an ellipse but I find it... | Let
\begin{equation}
x=\sqrt{\frac{2}{3}}\sin(t)
\end{equation}
\begin{equation}
y=\sqrt{\frac{2}{3}}\sin\left(t+\frac{2\pi}{3}\right)
\end{equation}
\begin{equation}
z=\sqrt{\frac{2}{3}}\sin\left(t-\frac{2\pi}{3}\right)
\end{equation}
Then $x+y+z=0$ and $x^2+y^2+z^2=1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1788368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Integrate $ \int \frac{1}{1 + x^3}dx $ $$ \int \frac{1}{1 + x^3}dx $$
Attempt:
I added and subtracted $x^3$ in the numerator but after a little solving I can't get through.
| Here is a roundabout way.
Let $$I:=\int\frac1{1+x^3}dx, J:=\int\frac x{1+x^3}dx,$$
then
$$I+J=\int\frac {1+x}{1+x^3}dx=\int\frac 1{1-x+x^2}dx=\int\frac1{\left(x-\frac1{2}\right)^2+\frac{3}{4}}dx=\frac2{\sqrt3}\arctan\frac{2x-1}{\sqrt3}+C,$$
$$I-J=\int\frac{1-x}{1+x^3}dx=\int\frac1{1+x}dx-\int\frac{x^2}{1+x^3}dx=\ln\lve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1788443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Limit of $\sqrt{\frac{\pi}{1-x}}-\sum\limits_{k=1}^\infty\frac{x^k}{\sqrt{k}}$ when $x\to 1^-$? I am trying to understand if
$$\sqrt{\frac{2\pi}{1-x}}-\sum\limits_{k=1}^\infty\frac{x^k}{\sqrt{k}}$$ is convergent for $x\to 1^-$. Any help?
Update: Given the insightful comments below, it is clear it is not converging, he... | $f(x):=\sum\limits_{k=1}^\infty\frac{x^k}{\sqrt{k}}$
$g(x):= \sqrt{\frac{2\pi}{1-x}}-f(x)$
$\sum\limits_{k=1}^\infty (-1)^{k-1} \frac{x^k}{\sqrt{k}}=f(x)-\sqrt{2}f(x^2)$ is convergent for $x\uparrow 1$ .
=>
$\sum\limits_{k=1}^\infty (-1)^k \frac{x^k}{\sqrt{k}}+\sqrt{\frac{2\pi}{1-x}}-\sqrt{2}\sqrt{\frac{2\pi}{1-x^2}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1788909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 2
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Characteristic function of Laplace distribution I'm trying to derive the characteristic function for the Laplace distribution with density $$\frac{1}{2}\exp\{-|x|\}$$
My attempt:
$$\frac{1}{2}\int_{\Omega}e^{itx-|x|}\mathrm{d}x$$
$$=\frac{1}{2}\int_0^\infty e^{(-it+1)-x}\mathrm{d}x+\frac{1}{2}\int_{-\infty}^0e^{(it+1)x... | First I will derive what we will need later using Euler's formula.
\begin{align*}
\int \cos(ux) e^{-x} dx + i \int \sin(ux) e^{-x} dx&= \int (\cos(ux) + i \cdot \sin(ux)) e^{-x} dx \\
&= \int e^{iux} \cdot e^{-x} dx \\
&= \int e^{(iu-1)x} dx \\
&= \frac{1}{iu-1} e^{(iu-1)x} \\
&= \frac{1}{iu-1} \frac{(iu+1)}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1791647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Complex numbers, finding solution for z How can you solve this? $z^2+2(1+i)z=2+2(\sqrt{3}-1)i$
I have tried to compare left and right side with real and imaginary part i then get
$ x^2+2x-y^2-2y=2$
$xy+x+y=(\sqrt{3}-1)$
But this equation can not be solved.
What else can i do? Setting in $z=re^{i\theta}$ does not help ... | solving
$$
z^2+2(1+i)z-2-2(\sqrt{3}-1)i=0
$$
we have:
$$
z=-1-i\pm\sqrt{(1+i)^2+2+2(\sqrt{3}-1)i}=-1-i\pm\sqrt{2}\sqrt{1+\sqrt{3}i}
$$
now:
$1+\sqrt{3}i=2e^{i\pi/3}$ and
$$
\sqrt{1+\sqrt{3}i}=\sqrt{2}e^{i\pi/6}=\sqrt{2}\left(\frac{\sqrt{3}}{2}+\frac{1}{2}i \right)
$$
so:
$$
z=-1-i\pm 2\left(\frac{\sqrt{3}}{2}+\frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1791733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
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Let $A= \{1,2,3,4,5,6,7,8,9,0,20,30,40,50\}$. 1. How many subsets of size 2 are there? 2.How many subsets are there altogether?
Let $A= \{1,2,3,4,5,6,7,8,9,0,20,30,40,50\}$.
1. How many subsets of size $2$ are there?
2.How many subsets are there altogether?
Answer:
1) I think there are $7$ subsets of size two a... | The comments have already very quickly pointed this out, but perhaps I can give an explanation on why they work.
$1$) André Nicolas' comment has already answered that there are $\binom{14}{2}$ subsets of size $2$. This notation is the binomial coefficient. That is, \begin{align}\binom{14}{2} &= \frac{14!}{(14-2)!2!}\\ ... | {
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"url": "https://math.stackexchange.com/questions/1795982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Limit of a Riemann sum: $\lim_{n\to\infty} {n^5 \sum^n_{r=0}\frac1{(n^2+r^2)^3}} $ Required to find
$\lim_{n\to{\infty}} {n^5 \sum^n_{r=0}\frac{1}{(n^2+r^2)^3}} $
$\lim_{n\to{\infty}} \frac{1}{n} \sum^n_{r=0}(\frac{n^2}{n^2+r^2})^3$
$\lim_{n\to{\infty}} \frac{1}{n} \sum^n_{r=0}(\frac{1}{1+\frac{r^2}{n^2}})^3$
$\lim_{n... | $$n^5\sum_{k=0}^n\frac1{(n^2+k^2)^3}=\frac1n\sum_{k=0}^n\frac1{\left(1+\left(\frac kn\right)^2\right)^3}\xrightarrow[n\to\infty]{}\int_0^1\frac{dx}{(1+x^2)^3}$$
Your substitution is fine...yet you must also change the limits accordingly:
$$x=\tan u\implies\begin{cases}x=0=\tan 0\implies u=0\\{}\\x=1=\tan u\implies u=\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1797653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Best way to expand $(2+x-x^2)^6$
I've completed part $(a)$ and gotten:
$64+192y+240y^2+160y^3+...$
Using intuition I substituted $x-x^2$ for $y$ and started listing the values for :
$y, y^2 $ and $y^3,$ in terms of $x$.
$y=(x-x^2)\\y^2=(x-x^2)^2 = x^2-2x^3+x^4;\\y^3 = (x-x^2)^3 = (x-x^2)(x^2-2x^3+x^4) = \;...$
Everyth... | I think I would say
$((x+2)(x-1))^6 = (x+2)^6(x-1)^6\\
(64 + 192x + 240x^2 + 160x^3 ...)(1 - 6x + 15x^2 - 20x^3 ...)$
I don't care about any powers bigger than 3
$64 + (64(-6) + 192)x + (64*15+192(-6) + 240)x^2 + (64(-20)+ 192*15 + 240*(-6) + 160)x^3...\\
64 - 192 x + 48 x^2 + 320x^3...$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1798501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Find the value of $\frac{a^2}{a^4+a^2+1}$ if $\frac{a}{a^2+a+1}=\frac{1}{6}$ Is there an easy to solve the problem? The way I did it is to find the value of $a$ from the second expression and then use it to find the value of the first expression. I believe there must be an simple and elegant approach to tackle the prob... | You are asked to express $\dfrac1B=\dfrac1{a^2+1+a^{-2}}$ in terms of $\dfrac1A=\dfrac1{a+1+a^{-1}}$.
Squaring "to see",
$$A^2=(a+1+a^{-1})^2=a^2+1+a^{-2}+2a+2+2a^{-1}=B+2A.$$
This gives us
$$B=A^2-2A=6^2-2\cdot6=24.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1798825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 1
} |
If $x, y, z$ are the side lengths of a triangle, prove that $x^2 + y^2 + z^2 < 2(xy + yz + xz)$
Question: If $x, y, z$ are the side lengths of a triangle, prove that $x^2 + y^2 + z^2 < 2(xy + yz + xz)$
My solution: Consider
$$x^2 + y^2 + z^2 < 2(xy + yz + xz)$$
Notice that $x^2+y^2+z^2=(x+y+z)^2-2(xy+yz+xz)$
Hence... | Expanding E.Girgin's answer and putting $x=b+c,y=a+c,z=a+b$ the original inequality turns into:
$$ 2a^2+2b^2+2c^2+2ab+2ac+2bc < 2(a^2+b^2+c^2+3ab+3ac+3bc) $$
that is pretty trivial.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1804220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Prove $(x+y)(y^2+z^2)(z^3+x^3) < \frac92$ for $x+y+z=2$ $x,y,z \geqslant 0$ and $x+y+z=2$, Prove
$$(x+y)(y^2+z^2)(z^3+x^3) < \frac92$$
While numerical method can solve this problem, I am more interested in classical solutions. I tried this problem for the past few months, using all kinds of AM-GM and CS, but still cann... | The proof comes by considering three cases.
Case 1: $z>1$, $y> 0.1$. We have
$$(x+y)(y^2+z^2)(z^3+x^3) - \frac92= (2-z)((2-x-z)^2+z^2)(z^3+(2-z-y)^3) - \frac92\\
< (2-z)((2-z)^2+z^2)(z^3+(2-z-0.1)^3) - \frac92$$
which is less than 0, see this plot:
and in more detail here:
Case 2: $z>1$, $y\leq 0.1$. We have
$$(x+y)(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1804897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 2,
"answer_id": 1
} |
Parametric equations for intersection between plane and circle So I was looking at this question Determine Circle of Intersection of Plane and Sphere
but I need to know how to find a parametric equation for intersections such as these. My particular question is to find a parametric equation for the intersection betwee... | First, let's name some points. Let $O=(0,0,0)$, $X=(1,0,0)$, $Y=(0,1,0)$, and $Z=(0,0,1)$. Let $A$ be the center of the circle we are trying to find. Let $K$ be the midpoint of $XY$. Note that $K=\left(\frac{1}{2},\frac{1}{2},0\right)$.
Let's consider right triangle $ZOK$:
We note that A is the base of the altitude t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1805161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
How to evaluate $\int_{0}^{1} \frac{\ln x}{x+1} dx$ I want to evaluate:
$$\int_{0}^{1} \frac{\ln x}{x+1} dx$$
If I was asked I would to evaluate:
$$\int_{0}^{1} \frac{\ln x}{x-1} dx$$
That would be easy because if I use the Taylor series for $\ln x$ centered at $1$ then things will cancel out and leave me with a easy i... | Consider:
$$I=\int_{0}^{1}\frac{\ln(x)}{1+x}dx= \sum_{n=1}^{\infty}\frac{(-1)^n}{n^2}$$
You can show this result by noticing since $0<x<1$ that $$\frac{1}{1+x}= \sum_{n=0}^{\infty}(-x)^n$$ and performing term by term integration. You will need integration by parts to do that.
Now you can also show that:
$$-I=\int_{0}^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1805230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A curious approximation to $\cos (\alpha/3)$ The following curious approximation
$\cos\left ( \frac{\alpha}{3} \right ) \approx \frac{1}{2}\sqrt{\frac{2\cos\alpha}{\sqrt{\cos\alpha+3}}+3}$
is accurate for an angle $\alpha$ between $0^\circ$ and $120^\circ$
In fact, for $\alpha = 90^\circ$, the result is exact.
How can ... | Let $y = \cos \alpha$ and $x = \cos(\alpha/3)$ then we know that
$$y = x(4x^{2} - 3)\tag{1}$$
Your approximation says that
$$x \approx \frac{1}{2}\sqrt{\frac{2y}{\sqrt{y + 3}} + 3}\tag{2}$$
or using $(1)$ we get $$\frac{y}{x} = 4x^{2} - 3 \approx \frac{2y}{\sqrt{y + 3}}\tag{3}$$ Canceling $y$ we get $$\frac{1}{x}\appro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1805643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to find an upper triangular with $\ U^2 = I $ which gives $\ U $ is its own inverse It's obvious that $\ I $(identity matrix) of any size $\ N $ satisfies$\ I^2 = I$ so that $\ I $ is its own inverse.
However if we consider $\ N = 2$ and attempt to find such a triangular matrix $\ U \neq I $ we have this scenario... | Here's one possible form. Let $a$ be any number. Define an $n \times n$ matrix as given below.
\begin{equation*}
A = \begin{bmatrix}
1 & a & \dfrac{a^2}{2} & \dfrac{a^3}{4}& \cdots & \dfrac{a^{n-1}}{2^{n-2}}\\
0 & -1 & -a & -\dfrac{a^2}{2} & \cdots & -\dfrac{a^{n-2}}{2^{n-3}}\\
0 & 0 & 1 & a & \cdots & \dfrac{a^{n-3}}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1806853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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I'm stuck in a logarithm question: $4^{y+3x} = 64$ and $\log_x(x+12)- 3 \log_x4= -1$ If $4^{y+3x} = 64$ and $\log_x(x+12)- 3 \log_x4= -1$ so $x + 2y= ?$
I've tried this far, and I'm stuck
$$\begin{align}4^{y+3x}&= 64 \\
4^{y+3x} &= 4^3 \\
y+3x &= 3 \end{align}$$
$$\begin{align}\log_x (x+12)- 3 \log_x 4 &= -1 \\
\log_x ... | Now you have
\begin{equation}
\log_x\left(\frac{x+12}{64}\right)=-1
\end{equation}
Therefore
\begin{equation}
\frac{x+12}{64}=x^{-1}
\end{equation}
Which leads to
\begin{equation}
x^2-12x-64=0
\end{equation}
Which can be factored
\begin{equation}
(x+16)(x-4)=0
\end{equation}
But of course $x$ cannot equal $-16$ so it... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1807717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
What to do when the integrating factor is a function of both x and y? I have to solve the following differential equation:
$$(\cos^2x + y \sin 2x) \frac{dy}{dx} + y^2 =0$$
using an integrating factor. An integrating factor that is a function of just $x$ or just $y$ won't work, so we need to find an integrating factor w... | Let's rewrite the equation in this form:
\begin{equation*}
\underbrace{y^2}_M \,\mathrm dx + \underbrace{(\cos^2 x + y \sin 2x)}_N \,\mathrm dy = 0.
\end{equation*}
Let $u(x,y)$ be an integrating factor. Then after multiplying the above equation by $u$, the resulting equation $uM \,\mathrm dx + uN \,\mathrm dy = 0$ sho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1809964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Calculating $\lim \limits_{x \to \infty} \frac{x+\frac12\cos x}{x-\frac12\sin x}$ using the sandwich theorem Calculating $\lim \limits_{x \to \infty} \dfrac{x+\frac12\cos x}{x-\frac12\sin x}$
Correct me if I'm wrong:
$\cos x$ and $\sin x$ are bounded so that
$$|\cos x|\le 1,\qquad |\sin x|\le1$$
Therefore I can say:
$... | Just to make more explicit the steps, we can assume $x>1$, so $x-\frac{1}{2}\sin x>0$ and $x+\frac{1}{2}\cos x>0$.
From $|\sin x|<1$, we get
$$
x-\frac{1}{2}\le x-\frac{1}{2}\sin x\le x+\frac{1}{2}
$$
Similarly,
$$
x-\frac{1}{2}\le x+\frac{1}{2}\cos x\le x+\frac{1}{2}
$$
Since all terms are positive, from
$$
x-\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1812388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Condition on $a$ for $(x^2+x)^2+a(x^2+x)+4=0$ Find the set of values of $a$ if $$(x^2+x)^2+a(x^2+x)+4=0$$ has
$(i)$ All four real and distinct roots
$(ii)$ Four roots in which only two roots are real and distinct.
$(iii)$ All four imaginary roots
$(iv)$ Four real roots in which only two are equal.
Now if I set $x^2+x=t... | Suppose we want to find the values of $a$ such that there are four real and distinct roots (values of $x$ such that the equation is true).
This means that if we let $x^2+x=t$ as you did, then we want the two possible values of $t$ to be distinct. Call them $t_1$ and $t_2$. We want that the following two equations both... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1814099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Show that: $97|2^{48}-1$ Show that: $97|2^{48}-1$
My work:
$$\begin{align}
2^{96}&\equiv{1}\pmod{97}\\
\implies (2^{48}-1)(2^{48}+1)&=97k\\
\implies (2^{24}-1)(2^{24}+1)(2^{48}+1) &=97k\\
\implies (2^{12}-1)(2^{12}+1)(2^{24}+1)(2^{48}+1)&=97k\\
\implies (2^6-1)(2^6+1)(2^{12}+1)(2^{24}+1)(2^{48}+1) &=97k
\end{align}$... | An elementary method follows:
Let $S$ denote the set $\{1,\cdots,48\}.$ Then every integer $n$ satisfies $n\equiv\pm k\pmod{97}$ for some $k\in S$ and a suitable choice of the sign. Now we consider $2k$ for $k\in S.$
For $k=1,\cdots,24,$ it is clear that $2k\in S$ so that we can take $2k\equiv l\pmod{97}$ for $l=2k.$
F... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1814416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Find the rightmost digit of: $1^n+2^n+3^n+4^n+5^n+6^n+7^n+8^n+9^n$ Find the rightmost digit of: $1^n+2^n+3^n+4^n+5^n+6^n+7^n+8^n+9^n(n$ arbitrary positive integer)
First of all I checked a few cases for small $n$'s and in all cases the rightmost digit was $5$, so maybe this is the case for all values of $n$.
Then I tho... | For $X>0$, and sequence indexed by $n > 0$:
$X^n$ repeats each $2 - 1$ in $\pmod 2$ since $2$ is prime.
$X^n$ repeats each $5 - 1$ in $\pmod 5$ since $5$ is prime.
So the sequence $X^n$ repeats each ${\rm gcd}(2 - 1, 5 - 1)$ in $\pmod {10}$, so you only have to check the value of
$$\sum_{X = 1}^9 X^n \pmod {10}$$
for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1815352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 5
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Find the center of the circle through the points $(-1,0,0),(0,2,0),(0,0,3).$ Find the center of the circle through the points $(-1,0,0),(0,2,0),(0,0,3).$
Let the circle passes through the sphere $x^2+y^2+z^2+2ux+2vy+2wz+d=0$ and the plane $Ax+By+Cz+D=0$
So the equation of the circle is $x^2+y^2+z^2+2ux+2vy+2wz+d+\lamb... | You essentially want to find the circumcenter $D$ of the triangle $\triangle_{ABC}$ with vertices
$$A = (-1,0,0), B = (0,2,0), C = (0,0,3)$$
Since $D$ is lying on the plane holding $A, B, C$, there exists $3$ real numbers
$\alpha,\beta,\gamma$ such that
$$D = \alpha A + \beta B + \gamma C\quad\text{ with }\quad \alpha ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1815547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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Prove by induction $\frac{7}{8}+7\left(\frac{1}{8}\right)^2+...+7\left(\frac{1}{8}\right)^n=1-\frac{1}{8^n}$ for every $n \in \mathbb N$
$\frac{7}{8}+7\left(\frac{1}{8}\right)^2+...+7\left(\frac{1}{8}\right)^n=1-\frac{1}{8^n}$ for every $n \in \mathbb N$
I first set $n=k$:
$\frac{7}{8}+7\left(\frac{1}{8}\right)^2+...... | First comment: you should begin by proving the base case, i.e. for $n=1$.
That being said, for the inductive step:
$$
1-\frac{1}{8^k}+7\frac{1}{8^{k+1}}
=
1-\frac{1}{8^k}+\frac{7}{8}\frac{1}{8^{k}}
=
1-\frac{8}{8}\frac{1}{8^k}+\frac{7}{8}\frac{1}{8^{k}}=
1-\frac{1}{8}\frac{1}{8^k}
=1-\frac{1}{8^{k+1}}
$$
and you can co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1816590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Given that $\tan 2x+\tan x=0$, show that $\tan x=0$ Given that $\tan 2x+\tan x=0$, show that $\tan x=0$
Using the Trigonometric Addition Formulae,
\begin{align}
\tan 2x & = \frac{2\tan x}{1-\tan ^2 x} \\
\Rightarrow \frac{2\tan x}{1-\tan ^2 x}+\tan x & = 0 \\
\ 2\tan x+\tan x(1-\tan ^2 x) & = 0 \\
2+1-\tan ^2 x & = 0 ... | $$\tan x+\tan2x=\dfrac{\sin(x+2x)}{\cos x\cos2x}$$
So, we need $\sin3x=0\implies3x=n\pi$ where $n$ is any integer
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1817022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
evaluating the integral $\int \frac{e^x(x^4+2)dx}{(x^2+1)^{5/2}}$ It accidentally came out like this but I need a fine proof for it.
$$\int e^x\frac{d}{dx}\left(\ln\left(x+\sqrt{x^2+1}\right)+\dfrac{x}{(x^2+1)^{3/2}}\right) dx$$
I am unable to figure out why $$\int\dfrac{1-2x^2}{\left(1+x^2\right)^{5/2}}dx =\dfrac{x}{\... | So you want to find a way to show:
$$\int\frac{1-2x^2}{\left(1+x^2\right)^{5/2}} \,\mbox{d}x =\dfrac{x}{\left(x^2+1\right)^{3/2}}$$
Let $x = \tan t$, then:
$$\int\dfrac{1-2x^2}{\left(1+x^2\right)^{5/2}} \,\mbox{d}x \to \int\dfrac{1-2\tan^2t}{\left(1+\tan^2t\right)^{5/2}} \sec^2 t \,\mbox{d}t$$
Now use $1+\tan^2 = \se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1819760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Prove $\int_0^{\infty} \frac{x^2}{\cosh^2 (x^2)} dx=\frac{\sqrt{2}-2}{4} \sqrt{\pi}~ \zeta \left( \frac{1}{2} \right)$ Wolfram Alpha evaluates this integral numerically as
$$\int_0^{\infty} \frac{x^2}{\cosh^2 (x^2)} dx=0.379064 \dots$$
Its value is apparently
$$\frac{\sqrt{2}-2}{4} \sqrt{\pi}~ \zeta \left( \frac{1}{2}... | $$I=\frac{1}{2\sqrt{2}}\int_{0}^{+\infty}\frac{\sqrt{u}\,du}{1+\cosh(u)}=\frac{1}{\sqrt{2}}\int_{1}^{+\infty}\frac{\sqrt{\log v}}{(v+1)^2}\,dv=\frac{1}{\sqrt{2}}\int_{0}^{1}\frac{\sqrt{-\log v}}{(1+v)^2}\,dv \tag{1}$$
but since
$$ \int_{0}^{1}v^k \sqrt{-\log v}\,dv = \frac{\sqrt{\pi}}{2(1+k)^{3/2}} \tag{2}$$
by expandi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1820280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 0
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$(\sin^{-1} x)+ (\cos^{-1} x)^3$ How do I find the least and maximum value of $(\sin^{-1} x)+ (\cos^{-1} x)^3$ ?
I have tried the formula $(a+b)^3=a^3 + b^3 +3ab(a+b)$ , but seem to reach nowhere near ?
| If $a+b=k,$
Method $\#1:$
$$a^3+b^3=(a+b)^3-3ab(a+b)=k^3-3kab$$
Now $$(a+b)^2-4ab=(a-b)^2\ge0\iff-4ab\ge-(a+b)^2$$
Method $\#2:$
$$a^3+b^3=a^3+(k-a)^3=k^3-3k^2a+3ka^2=k^3+3k\left(a^2-ka\right)$$
Now $a^2-ka=\dfrac{(2a-k)^2-k^2}4\ge-\dfrac{k^2}4$
For both methods, here $$k=\dfrac\pi2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1821252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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If $U^*DU=D=V^*DV$ for diagonal $D$, is $U^*DV$ diagonal too? All the matrices mentioned are complex $n\times n$ matrices. Let $U, V$ be unitary matrices such that $U^*DU=V^*DV=D$ for a diagonal matrix $D$ with nonnegative diagonal entries. Does this imply that $U^*DV$ is also diagonal? All I understand is that $U^*V$ ... | In general, no. Consider the permutation matrices
$$
E_1 \;\; =\;\; \left [ \begin{array}{ccc}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1 \\
\end{array} \right ] \;\;\;\;\; E_2 \;\; =\;\; \left [ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0 \\
\end{array} \right ].
$$
These will diagonalize any diagonal matrix $D... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1821503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Use integration by parts $\int^{\infty}_{0} \frac{x \cdot \ln x}{(1+x^2)^2}dx$ $$I=\int^{\infty}_{0} \frac{x \cdot \ln x}{(1+x^2)^2}dx$$
Clearly $$-2I=\int^{\infty}_{0} \ln x \cdot \frac{-2x }{(1+x^2)^2} dx$$
My attempt :
$$-2I=\left[ \ln x \cdot \left(\frac{1}{1+x^2}\right)\right]^\infty_0 - \int^{\infty}_{0} \left(\f... |
Integration by parts works fine provided one writes the integral $I$ as
$$I=\lim_{\epsilon\to 0^+}\lim_{L\to \infty}\int_{\epsilon}^L\frac{x\log(x)}{(1+x^2)^2}\,dx$$
Then, proceeding with integration by parts, we find
$$\begin{align}
I&=\frac14\lim_{\epsilon\to 0^+}\lim_{L\to \infty}\left.\left(\frac{2x^2\log(x)}{1+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1822013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
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Why my calculations aren't right? (Maclaurin series) Good evening to everyone!
I tried to calculate $ \cos\left( x- \frac{x^3}{3} + o(x^4)\right) $ using the MacLaurin series but instead of getting the final result equal to $1 - \frac{x^2}{2}+\frac{3x^4}{8} + o(x^4)$ I got this:
$$
\cos\left( x- \frac{x^3}{3} + o(x^4)\... | In the first line you should consider the third part of MacLaurin series of $cos(u)$ which is $+\frac{u^4}{4!}$. This way, you would get the right result.
In fact, by now, your first line is not correct because there are some coefficient of $x^4$ in $\frac{(x-\frac{x^3}{3}+o(x^4))^4}{4!}$.
Moreover the second term of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1822220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Finding the second derivative of $f(x) = \frac{4x}{x^2-4}$. What is the second derivative of
$$f(x) = \frac{4x}{x^2-4}?$$ I have tried to use the quotient rule but I can't seem to get the answer.
| To simplify the differentiation, we can first rewrite the function as a sum of partial fractions. To do this, we assert that $\frac{4x}{x^2 - 4}$ can be written as $\frac{A}{x + 2} + \frac{B}{x - 2},$ with $A$ and $B$ constants. If this is the case, then clearly, $(A + B)x + 2(B - A) = 4x,$ from which we conclude that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1822715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3} +...+ \frac{n}{2^n} = 2 - \frac{n+2}{2^n} $ I need help with this exercise from the book What is mathematics? An Elementary Approach to Ideas and Methods. Basically I need to proove:
$$\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3} +...+ \frac{n}{2^n} = 2 - \frac{n+2}{2^n} $... | In $P2$ you have made a misktake. It will be, $$\dfrac{2^{k+1}-k\color{red}{-}2}{2^k}+\dfrac{k+1}{2^{k+1}}=\dfrac{2^{k+2}-k\color{red}{-}3}{2^{k+1}}$$
Now if you simplify LHS then,
$$\dfrac{2^{k+2}-2k-4+k+1}{2^{k+1}}=\dfrac{2^{k+2}-k-3}{2^{k+1}}=2-\dfrac{k+3}{2^{k+1}}$$and you are done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1822879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solving integral without partial fraction I would like to expand my pool of integral solving skills and thus try to solve older problems again, however with a different method I had used back then, when I encountered them first. For this problem I omitted the lower and upper bound and just compute the indefinite integr... | Be smart, add zeros!
$$I=\int dx\frac{1}{x^4-1}=\\
\int dx\frac{1+\overbrace{x^2-x^2}^{=0}}{(x^2-1)(x^2+1)}=\\\int dx\frac{1}{x^2-1}-\int dx\frac{x^2}{(x^2-1)(x^2+1)}=\\
\int dx\frac{1}{x^2-1}-\int dx\frac{x^2\overbrace{-1+1}^{=0}}{(x^2-1)(x^2+1)}=\\
\int dx\frac{1}{x^2-1}-\int dx\frac{1}{x^2+1}-\underbrace{\int dx\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1824690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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If $\sin x + \csc x =2 \tan x$. Find value of $\cos^9x +\cot^9x +\sin^7x$ Problem:
If $\sin x+\csc x=2\tan x$, Find value of $\cos^9x+\cot^9x+\sin^7x$
Solution:
\begin{align*}&\sin x+\csc x=2\tan x \\ &\sin x+\frac{1}{\sin x}=2\frac{\sin x}{\cos x} \\ &\sin^2x+1=2\frac{\sin^2x}{\cos x} \\ &\sin^2x\cos x+\cos x=2\si... | This is not really an answer, but a comment with image indicating that something is wrong with the problem in case the options are $1$, $0$, $-1$ and $2$ as stated in a comment above.
In the picture below I have let Mathematica draw the graphs of $\sin x+\csc x$ (blue), $2\tan x$ (yellow), $\cos^9x+\cot^9x+\sin^7x$ (gr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1825376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Prove that $ 1+2q+3q^2+...+nq^{n-1} = \frac{1-(n+1)q^n+nq^{n+1}}{(1-q)^2} $ Prove:
$$ 1+2q+3q^2+...+nq^{n-1} = \frac{1-(n+1)q^n+nq^{n+1}}{(1-q)^2} $$
Hypothesis:
$$ F(x) = 1+2q+3q^2+...+xq^{x-1} = \frac{1-(x+1)q^x+xq^{x+1}}{(1-q)^2} $$
Proof:
$$ P1 | F(x) = \frac{1-(x+1)q^x+xq^{x+1}}{(1-q)^2} + (x+1)q^x = \frac{1-(x... | Here is an alternative approach, motivated by the fact that it's usually wise in such problems to multiply the $(1-q)$ factors through:
\begin{align}
(1-q)^2(1+2q+3q^2+\cdots +nq^{n-1})
&=(1-q)\cdot (1-q)(1+2q+3q^2+\cdots +nq^{n-1})\\
&=(1-q)\cdot (1+q+q^2+\cdots +q^{n-1}-nq^{n})\\
&=1-(n+1)q^{n}+nq^{n+1}.
\end{align}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1825825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$\int_0^\infty \frac{ x^{1/3}}{(x+a)(x+b)} dx $ $$\int_0^\infty \frac{ x^{1/3}}{(x+a)(x+b)} dx$$ where $a>b>0$
What shall I do?
I have diffucty when I meet multi value function.
| If we apply the substitution $x=y^3$ we get:
$$ I(a,b) = 3\int_{0}^{+\infty}\frac{y^3\,dy}{(y^3+a)(y^3+b)}=\frac{3}{a-b}\int_{0}^{+\infty}\frac{a\,dy}{y^3+a}-\frac{3}{a-b}\int_{0}^{+\infty}\frac{b\,dy}{y^3+b}$$
but if we set, for any $c>0$,
$$ J(c) = \int_{0}^{+\infty}\frac{c\,dy}{y^3+c},$$
we simply have $J(c)= c^{1/3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1826607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Find stationary points of the function $f(x,y) = (y^2-x^4)(x^2+y^2-20)$ I have problem in finding some of the stationary points of the function above. I proceeded in this way: the gradient of the function is:
$$ \nabla f = \left( xy^2-3x^5-2x^3y^2+40x^3 ; x^2y+2y^3-x^4y-20y \right) $$
So in order to find the stationary... | As you implied, you can factor the original system as $x(3x^4-40x^2+2x^2y^2-y^2)$ and $y(x^4-x^2-2y^2+20)=0$.
Taking $y=0$ gives you $x=0,x=\pm2\sqrt{\frac{10}{3}}$. Taking $x=0$ gives you $y=\pm\sqrt{10}$.
So you are left to solve $3x^4-40x^2+2x^2y^2-y^2=0,x^4-x^2-2y^2+20=0$ and $x,y\ne0$. Substituting from the secon... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show that $5^n$ divides $F_{5^n}$. If $F_n$ denotes the $n$-th Fibonacci number ($F_0 = 0, F_1 = 1, F_{n+2} = F_{n+1} + F_n$), show that $5^n$ divides $F_{5^n}$.
| Note that
$$
\begin{pmatrix}a+b & a \\ a & b\end{pmatrix}^5
=
\begin{pmatrix}* & c \\ c & *\end{pmatrix}
$$
where $c=5 a (a^4 + 3 a^3 b + 4 a^2 b^2 + 2 a b^3 + b^4)$. Therefore, if $5^n$ divides $a$, then $5^{n+1}$ divides $c$.
Apply this to
$$
\begin{pmatrix}1&1\\1&0\end{pmatrix}^n
=
\begin{pmatrix}F_{n+1}&F_n\\F_n&F_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1827074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving that $3 + 3 \times 5 + 3 \times 5^2 + \cdots+ 3 \times 5^n = [3(5^{n+1} - 1)] / 4$ whenever $n \geq 0$
Use induction to show that $$3 + 3 \times 5 + 3 \times 5^2 + \cdots+ 3 \times 5^n= \frac{3(5^{n+1} - 1)}{4} $$whenever $n$ is a non-negative integer.
I know I need a base-case where $n = 0$:
$$3 \times 5^0 =... | You are trying to show that
$$\sum_{n=0}^{N}5^n=\frac{5^{N+1}-1}{4}$$
We can leave out the factor of $3$ since it just multiplies both sides. The base case is simple, you just have $1=1$. Now assume it is true for $N$. Then we have
$$\sum_{n=0}^{N+1}5^n=\sum_{n=0}^{N}5^n+5^{N+1}=\frac{5^{N+1}-1}{4}+5^{N+1}=\frac{5^{N+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1828029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the value of $\frac{a^2}{b+c} + \frac{b^2}{a+c} + \frac{c^2}{a+b}$ if $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 1$? If $$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 1$$ then find the values of $$\frac{a^2}{b+c} + \frac{b^2}{a+c} + \frac{c^2}{a+b}.$$ How can I solve it? Please help me. Thank you in ad... | You have $$a+b+c = (a+b+c)\left(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\right) = \frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b} + a+b+c.$$
Then you get $$\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b} = 0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1828288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Prove that $\int_{0}^{\infty}{1\over x^4+x^2+1}dx=\int_{0}^{\infty}{1\over x^8+x^4+1}dx$ Let
$$I=\int_{0}^{\infty}{1\over x^4+x^2+1}dx\tag1$$
$$J=\int_{0}^{\infty}{1\over x^8+x^4+1}dx\tag2$$
Prove that $I=J={\pi \over 2\sqrt3}$
Sub: $x=\tan{u}\rightarrow dx=\sec^2{u}du$
$x=\infty \rightarrow u={\pi\over 2}$, $x=0\... | For $\theta$ an arbitrary constant, we have
\begin{equation}
x^4+2x^2\cos2\theta+1=(x^2-2x\sin\theta+1)(x^2+2x\sin\theta+1)\tag1
\end{equation}
Now, let’s evaluate
\begin{equation}
I=\int_0^\infty\frac{1}{x^4+2x^2\cos2\theta+1}\ dx\tag2
\end{equation}
By making substitution $x\mapsto\frac1x$, then
\begin{equation}
I=\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1829298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 9,
"answer_id": 6
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$5^{th}$ degree polynomial expression
$p(x)$ is a $5$ degree polynomial such that
$p(1)=1,p(2)=1,p(3)=2,p(4)=3,p(5)=5,p(6)=8,$ then $p(7)$
$\bf{My\; Try::}$ Here We can not write the given polynomial as $p(x)=x$
and $p(x)=ax^5+bx^4+cx^3+dx^2+ex+f$ for a very complex system of equation,
plz hel me how can i solve that... | Let's do it in the most elementary way. Let $$Q(x)=P(x+1)-P(x)-x+2 \tag{1}$$Observe that $Q$ is of degree $4$ and $Q(3)=Q(4)=Q(5)=0$. Therefore we can write$$Q(x)=a(x-3)(x-4)(x-5)(x-b) \tag{2}$$You have also from $(1)$ that $Q(1)=Q(2)=1$, which after substitution in $(2)$ you get $a=-1/8$ and $b=2/3$. So $$Q(6)=-\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1832885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the value of $ [1/ 3] + [2/ 3] + [4/3] + [8/3] +\cdots+ [2^{100} / 3]$ Assume that [x] is the floor function. I am not able to find any patterns in the numbers obtained. Any suggestions?
$$[1/ 3] + [2/ 3] + [4/3] + [8/3] +\cdots+ [2^{100} / 3]$$
| Starting from $n=0$, even-$n =2k$ terms are $(4^k - 1)/3$ and the subsequent odd-$n = 2k+1$ terms are $2\times (4^k- 1)/3$. So overall:
\begin{align}
\sum_{k=0}^{49}\left[(1+2)\frac{4^{k} - 1}{3}\right] + \frac{4^{50}-1}{3} &= \sum_{k=0}^{49}(4^{k} - 1) + \frac{4^{50}-1}{3} \\
& = \sum_{k=0}^{49}(4^{k}) + \frac{4^{50}-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1833916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Find $y(2) $ given $y(x)$ given a separable differential equation Find what $y(2)$ equals if $y$ is a function of $x$ which satisfies:
$x y^5\cdot y'=1$ given $y=6$ when $x=1$
I got $y(2)=\sqrt{6\ln(2)-46656}$
but this answer is wrong can anyone help me figure out the right answer and how I went wrong?
| $$
x \, y^5\, y' = 1 \Rightarrow \\
\int y^5 \, dy = \int \frac{dx}{x} \Rightarrow \\
\frac{1}{6} y^6 = \ln(x) + C \quad (x > 0)
$$
Inserting $y(1) = 6$ gives
$$
\frac{1}{6} 6^6 = \ln(1) + C \Rightarrow \\
C = 6^5
$$
so we got the solution:
$$
y^6 = 6 \ln(x) + 6^6 \quad (x > 0)
$$
which yields
$$
y(2)^6 = 6 \ln(2) + 6... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1834796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Taking Mod on both sides, mathematically correct? When given a equation containing complex numbers such as
$$ \frac{a+ib}{c+id} = x + iy$$
and required to prove
$$ \frac{a^2 +b^2}{c^2+d^2} = x^2 + y^2$$
Is taking the mod of both sides a legal mathematical step? I ask so because my textbook first finds the conjugate ... | Yes, taking the mod of both sides is mathematically valid, but we don't necessarily need to do that.
Convert to polar:
$$\frac{\sqrt{a^2+b^2}\text{cis}(\theta_1)}{\sqrt{c^2+d^2}\text{cis}(\theta_2)}=\sqrt{x^2+y^2}\text{cis}(\theta_3)$$
Separate the moduli and arguments:
$$\frac{\sqrt{a^2+b^2}}{\sqrt{c^2+d^2}\sqrt{x^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1834891",
"timestamp": "2023-03-29T00:00:00",
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Q23 from AMC 2012 If $abc+ab+bc+ac+a+b+c=104$, and $a,b,c>0$, what is the value of $ a^2 + b^2 +c^2$?
I tried to make $a,b,c$ the subject of the equation and tried to add them up but it's weird...
I also have lots of more questions which I will ask later from AMC 2012, it's more difficult than what I usually do...
| Given, $abc+ab+bc+ac+a+b+c=104$, and $a,b,c>0$
$abc+ab+bc+ac+a+b+c=104$
$(a+1)(b+1)(c+1)-1=104$
$(a+1)(b+1)(c+1)=105$
$(a+1)(b+1)(c+1)=3\cdot5\cdot7$
Without the loss of generality, let $a\leq b\leq c$
$a+1=3, b+1=5, c+1=7$
$a=2, b=4, c=6$
$\;\therefore\;a^2+b^2+c^2=2^2+4^2+6^2$
Hence, $a^2+b^2+c^2=56$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1835659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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I want to show that $\int_{-\infty}^{\infty}{\left(x^2-x+\pi\over x^4-x^2+1\right)^2}dx=\pi+\pi^2+\pi^3$ I want to show that
$$\int_{-\infty}^{\infty}{\left(x^2-x+\pi\over x^4-x^2+1\right)^2}dx=\pi+\pi^2+\pi^3$$
Expand $(x^4-x+\pi)^2=x^4-2x^3+2x^2-2x\pi+\pi{x^2}+\pi^2$
Let see (substitution of $y=x^2$)
$$\int_{-\infty}... | Evaluating the integral of interest can be reduced to evaluating the integral
$$I(a,b)=\int_0^\infty \frac{1}{x^4+bx^2+a} \,dx\tag 1$$
To see this, we exploit first odd symmetry to write the integral of interest as
$$\begin{align}
\int_{-\infty}^\infty \frac{(x^2-x+\pi)^2}{(x^4-x^2+1)^2}\,dx&=2\int_0^\infty \frac{x^4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1836306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 5,
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How to determine which of the following matrices are similar? If we have the following three matrices:
$$
A=\begin{bmatrix}
7 &1 \\
-5 &3
\end{bmatrix},\;\;
B=\begin{bmatrix}
5 &-1 \\
1 &5
\end{bmatrix},\;\;
C=\begin{bmatrix}
5 &1 \\
1 &5
\end{bmatrix}.
$$
What is the right procedure to determine if matrices are... | As similar matrices have similar determinants, matrix $C$ is not similar to $A$ or $B$ as $\det C = 24$ but $\det A = \det B = 26$.
Now we notice that $A$ and $B$ have the similar egeinvalues from characteristic equation
$$
\lambda^2 - 10\lambda + 26 = 0
$$
which gives
$$
\lambda_{1,2} = 5 \pm i.
$$
We may diagoniliz... | {
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"url": "https://math.stackexchange.com/questions/1836890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$x, y , z$ are respectively the $sines$ and $p, q, r$ are respectively the $cosines$ of the ..... $x, y , z$ are respectively the $sines$ and $p, q, r$ are respectively the $cosines$ of the angles $\alpha, \beta, \gamma$, which are in A.P. with common difference $\frac{2\pi}{3}$.
1. $yz + zx + xy = ?$
2. $x^2 (qy - r... | WLOG let $x=\sin\left(A-\dfrac{2\pi}3\right),y=\sin A,z=\sin\left(A+\dfrac{2\pi}3\right)$
$$2(xy+yz+zx)=(x+y+z)^2-(x^2+y^2+z^2)$$
We can prove $x+y+z=0$
Using $\cos2B=1-2\sin^2B,$
$$2(x^2+y^2+z^2)=3-\left\{\cos\left(2A-\dfrac{4\pi}3\right)+\cos2A+\cos\left(2A+\dfrac{4\pi}3\right)\right\}$$
Now $\cos\left(2A-\dfrac{4\p... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Let $P(x)=ax^{2014}-bx^{2015}+1$ and $Q(x)=x^2-2x+1$ such that $Q(x)|P(x)$, find $a+b$
Let $P(x)=ax^{2014}-bx^{2015}+1$ and $Q(x)=x^2-2x+1$ be the polynomials where $a$ and $b$ are real numbers. If polynomial $P$ is divisible by $Q$, what is the value of $a+b$.
This is what I have tried so far: Since $Q(x)|P(x)$ we h... | Note that $Q(x) = (x-1)^2$, which means that $(x-1) \mid P(x)$ and $(x-1) \mid P'(x)$. In other words $x=1$ is a zero of both $P(x)$ and $P'(x)$. So after all you're left to solve the system of linear equation:
$$\begin{cases} a-b+1 = 0 \\ 2014a - 2015b = 0 \end{cases}$$
It's easy to conclude that $a=-2015$ and $b=-201... | {
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"url": "https://math.stackexchange.com/questions/1838433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove $\frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$. So I have to prove
$$ \frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}.$$
I rearranged it
$$ a^2bc + ab^2c + abc^2 \leq b^2c^2 + a^2c^2 + a^2b^2 .$$
My idea from there is somehow using the AM-GM inequality. Not sure how t... | Write $x=1/a$, $y=1/b$ and $z=1/c$. We get $$xy+yz+zx\leq x^2+y^2+z^2$$ Now use that $$p^2+q^2\geq 2pq$$
for each pair $p,q\in \{x,y,z\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1840148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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How many ways a 9 digit number can be formed using the digits 1 t0 9 without repetition such that it is divisble by $11$. How many ways a 9 digit number can be formed using the digits 1 t0 9 without repetition such that it is divisible by $11$.
My attempt-
A number is divisible by 11 if the alternating sum of its digit... | $$1+2+...+9=\frac{9\cdot10}{2}=45\\(x_1+x_3+x_5+x_7+x_9)-(x_2+x_4+x_6+x_8)=11m$$
$m$ must be $1$ or $3$ since 45 is odd and $3$ is discarded immediately so we have
$$\begin{cases}X+Y=45\\X-Y=11\end{cases}$$ Hence $X=28$ and $Y=17$ We work with $28$.
1) With $9$ and $8$ one has $x_1+x_2=11$ and $x_1+x_2+x_3=11$ (becaus... | {
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"source": "stackexchange",
"question_score": "7",
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"answer_id": 3
} |
Is something wrong with this solution for $\sin 2x = \sin x$? I have this question. What are the solutions for $$
\sin 2x = \sin x; \\ 0 \le x < 2 \pi $$
My method:
$$ \sin 2x - \sin x = 0 $$
I apply the formula $$ \sin a - \sin b = 2\sin \left(\frac{a-b}{2} \right) \cos\left(\frac{a+b}{2} \right)$$
So:
$$ 2\sin\le... | There's nothing wrong up to the reduction to
$$
\sin\frac{x}{2}\cos\frac{3x}{2}=0
$$
Then you have either
$$
\sin\frac{x}{2}=0
$$
that is, $x/2=k\pi$ and $x=2k\pi$, or
$$
\cos\frac{3x}{2}=0
$$
so
$$
\frac{3x}{2}=\frac{\pi}{2}+k\pi
$$
and
$$
x=\frac{\pi}{3}+\frac{2k\pi}{3}
$$
Now let's examine the first set of solutions... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1845034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 1
} |
If $x\in \left(0,\frac{\pi}{4}\right)$ then $\frac{\cos x}{(\sin^2 x)(\cos x-\sin x)}>8$
If $\displaystyle x\in \left(0,\frac{\pi}{4}\right)\;,$ Then prove that $\displaystyle \frac{\cos x}{\sin^2 x(\cos x-\sin x)}>8$
$\bf{My\; Try::}$ Let $$f(x) = \frac{\cos x}{\sin^2 x(\cos x-\sin x)}=\frac{\sec^2 x}{\tan^2 x(1-\ta... | Since $\displaystyle 0<t<1, \;\frac{1+t^2}{t^2(1-t)}>8\iff 1+t^2>8t^2(1-t)=8t^2-8t^3\iff8t^3-7t^2+1>0$
If $g(t)=8t^3-7t^2+1,\;\; g^{\prime}(t)=24t^2-14t=0\iff t=0 \text{ or }t=\frac{7}{12}$.
Since $g(\frac{7}{12})=\frac{89}{432}$ is the minimum value of $g$ on $(0,1)$, $\;\;g(t)>0$ for $0<t<1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1846161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 1
} |
Genereating function of $H_{2n}$ We know the generating function of: $$\sum_{n=1}^{\infty}H_nx^n=\frac{\ln(1-x)}{x-1}$$.
How do we find out the generating function of $$\sum_{n=1}^{\infty}H_{2n}x^n$$
I used the formula: $\displaystyle { H }_{ 2n }=\frac { 1 }{ 2 } \left[ { H }_{ n }+{ H }_{ n-\frac { 1 }{ 2 } } \right... | Define $$f(x) = \sum_{n=1}^\infty H_{2n} x^{2n}, \quad g(x) = \sum_{n=1}^\infty H_{2n-1} x^{2n-1}.$$ Then since $$H_{2n} = H_{2n-1} + \frac{1}{2n},$$ we have $$f(x) = \sum_{n=1}^\infty H_{2n} x^{2n} = x \sum_{n=1}^\infty H_{2n-1} x^{2n-1} + \sum_{n=1}^\infty \frac{x^{2n}}{2n} = x g(x) - \frac{1}{2}\log(1-x^2).$$ But ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1846505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Need a solution to this Integration problem How to evaluate:$\displaystyle\int_{0}^{r}\frac{x^4}{(x^2+y^2)^{\frac{3}{2}}}dx$
I have tried substituting $x =y\tan\ A$, but failed.
| Hint
$$I=\displaystyle\int_{0}^{r}\frac{x^4}{(x^2+y^2)^{\frac{3}{2}}}dx=\displaystyle\int_{0}^{r}\frac{(x^2+y^2)^2}{(x^2+y^2)^{\frac{3}{2}}}dx-2\displaystyle\int_{0}^{r}\frac{x^2y^2}{(x^2+y^2)^{\frac{3}{2}}}dx-y^2\displaystyle\int_{0}^{r}\frac{1}{(x^2+y^2)^{\frac{3}{2}}}dx$$
$$I=\displaystyle\int_{0}^{r}\sqrt{x^2+y^2}d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1846579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Prove $\frac{a}{b} + \frac{b}{c}+\frac{c}{a} \geq \frac{c+a}{c+b} + \frac{a+b}{a+c} + \frac{b+c}{b+a}$ Prove that $\frac{a}{b} + \frac{b}{c}+\frac{c}{a} \geq \frac{c+a}{c+b} + \frac{a+b}{a+c} + \frac{b+c}{b+a}$ with a,b,c > 0
| Show ${a\over b} +{b\over c} +{c\over a} \ge {c+a\over c+b} + {a+b\over a+c} +{b+c\over b+a}$
But this is the same as (1) a/b+b/c+c/a >= (2) avg(c,a)/avg(c,b)+avg(a,b)/avg(a,c)+avg(b,c)/avg(b,a) . Thus it is sufficient to show a (mean preserving) spread of the inputs increases avg{a/b,b/c,c/a} . This is easily ve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1846697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What is the range of $λ$? Suppose $a, b, c$ are the sides of a triangle and no two of them are equal.
Let $λ ∈ IR$. If the roots of the equation $x^
2 + 2(a + b + c)x + 3λ(ab +
bc + ca) = 0$ are real, then what is the range of $λ$?
I got that $$λ ≤\frac{
(a + b + c)^
2}
{3(ab + bc + ca)}$$
After that what to do?
| There is a systematic way to get rid of the triangle condition by substituting
$$a = x + y, b = y + z, c = z + x$$
which comes from the tangents to the incircle, explained e.g. here. Now the only condition on $x,y,z$ is that they are positive reals, and additionally they are distinct (because we want $a,b,c$ to be dis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1848389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Values of $a$ for Which $|||x-1|-3|-a|=k$ Has 8 Distinct Real Roots Question
$|||x-1|-3|-a|=k$, $a \in \mathbb{N}$ has 8 distinct real roots for some $k$, then find the number of such values of $a$.
My Thought
Sorry I cannot show any work this time because I don't understand how to proceed. Please provide some guidance... | Of course, we have to assume that $k \ge 0$.
\begin{align}
|||x-1|-3|-a| &= k \\
||x-1|-3|-a &= \pm k \\
||x-1|-3| &= a \pm k \\
|x-1|-3 &= \pm(a \pm k) \\
|x-1| &= 3 \pm(a \pm k) \\
x-1 &= \pm(3 \pm(a \pm k)) \\
x &= 1 \pm(3 \pm(a \pm k)) \\
\end{al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1848960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Proving $ a^4 \equiv 1 \pmod d$ I need to prove the following statements:
Prove the following statements:
(a) if $a$ is odd then $a^4 ≡ 1 \pmod 4$,
(b) if $5$ does not divide a, then $a^4 \equiv 1 \pmod 5$.
Can I do this inductively? Or should I be adopting another approach? I know for (a), if $a$ is odd, $a^4$ will al... |
If $a$ is odd, then $a^4\equiv1\pmod4$:
*
*$a\equiv\color\red1\pmod4 \implies a^4\equiv\color\red1^4\equiv1\pmod4$
*$a\equiv\color\red3\pmod4 \implies a^4\equiv\color\red3^4\equiv81\equiv1\pmod4$
If $5$ does not divide $a$, then $a^4\equiv1\pmod5$:
*
*$a\equiv\color\red1\pmod5 \implies a^4\equiv\color\red... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1849821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Decreasing sequence numbers with first digit $9$
Find the sum of all positive integers whose digits (in base ten) form a strictly decreasing sequence with first digit $9$.
The method I thought of for solving this was very computational and it depended on a lot of casework. Is there a nicer way to solve this question?... | If the digit with value $10^j$ is $k$, there are $\binom kj$ options for the digits after that and $2^{8-k}$ options for the digits before that (corresponding to the subsets of the digits between $k$ and $9$), except for $k=9$ there is $1$ option for the digits before (namely none). Thus the sum of the contributions fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1851478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Find $\sin \theta $ in the equation $8\sin\theta = 4 + \cos\theta$ Find $\sin\theta$ in the following trigonometric equation
$8\sin\theta = 4 + \cos\theta$
My try ->
$8\sin\theta = 4 + \cos\theta$
[Squaring Both the Sides]
=> $64\sin^{2}\theta = 16 + 8\cos\theta + \cos^{2}\theta$
=> $64\sin^{2}\theta - \cos^{2}\theta= ... | Or you can use the half angle formulas:
$\cos(\theta) = \frac{1 - \tan^2(\frac{\theta}{2})}{1 + \tan^2(\frac{\theta}{2})} $
$\sin(\theta) = \frac{2 \tan(\frac{\theta}{2})}{1 + \tan^2(\frac{\theta}{2})}$
So as to get:
$ 8 \frac{2 \tan(\frac{\theta}{2})}{1 + \tan^2(\frac{\theta}{2})} = 4 + \frac{1 - \tan^2(\frac{\theta}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1853845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
To find the solution of the equation $2\left|z \right|-4az+1+ia=0$ Question:-
For every real number $a \ge 0$, find all the complex numbers $z$, satisfying the equation $2\left|z \right|-4az+1+ia=0$
Attempt at a solution:-
Let $z=x+iy$, then the equation $2\left|z \right|-4az+1+ia=0$ becomes as follows
$$\begin{equati... | For $a=0$, there is no solution. In the following, $a\gt 0$.
You have
$$2\sqrt{x^2+\dfrac{1}{16}}=4ax-1\tag2$$
Here, note that you need to have
$$4ax-1\gt 0\iff x\gt \frac{1}{4a}\tag3$$
Squaring the both sides of $(2)$ gives
$$(64a^2-16)x^2-32ax+3=0\tag4$$
Case 1 : For $64a^2-16=0$, i.e. $a=1/2$, $x=3/16$ but this does... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1853952",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.