Questions tagged [integration]
For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.
14,415
questions with no upvoted or accepted answers
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Are these generalizations known in the literature?
By using
$$\int_0^\infty\frac{\ln^{2n}(x)}{1+x^2}dx=|E_{2n}|\left(\frac{\pi}{2}\right)^{2n+1}\tag{a}$$
and
$$\text{Li}_{a}(-z)+(-1)^a\text{Li}_{a}(-1/z)=-2\sum_{k=0}^{\lfloor{a/2}\rfloor }\frac{\eta(...
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What is $\int_0^1 \left(\tfrac{\pi}2\,_2F_1\big(\tfrac13,\tfrac23,1,\,k^2\big)\right)^3 dk$?
As in this post, define the ff:
$$K_2(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)}$$
$$K_3(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac13,\tfrac23,1,\,k^2\right)}$$
$$K_4(k)={\tfrac{\...
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Theorem 6.17 in Baby Rudin, 3rd ed: $\int_a^b f \,d\alpha = \int_a^b f(x) \alpha^\prime(x) \,dx$
Here is Theorem 6.17 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:
Assume $\alpha$ increases monotonically and $\alpha^\prime \in \mathscr{R}$ on $[a, b]$. Let $f$ be ...
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24
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When is $\int_0^1 \int_0^1 \frac{f(x) - f(y)}{x-y} \, \text{d} x \, \text{d} y = 2 \int_0^1 f(t) \log\left(\frac{t}{1-t}\right) \, \mathrm{d} t$?
Double integrals of this type sometimes appear when using differentiation under the integral sign with respect to two variables. Therefore, I am interested in reducing them to (simpler) single ...
- 11.7k
23
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answer
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Supremum of $\int_{-\infty}^{\infty}\frac{p'(x)^2}{p(x)^2+p'(x)^2}\,\mathrm{d}x$
Question. Let $P_d = \{ p \in \mathbb{R}[x] : \deg p = d\}$ denote the set of all degree $d$ polynomials with real coefficients. Also, for $p \in \mathbb{R}[x]$, define
$$ I(p) = \frac{1}{\pi} \int_{-\...
- 170k
23
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0
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Difficult integral for a marginal distribution
I am trying to derive a marginal probability distribution for $y$, and failed, having tried all methods to solve the following integral:
$$
\operatorname{p}\left(y\right) =
\int_0^{1/\sqrt{\,2\pi\,}}\!...
- 3,769
21
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Prove $ \int_0^1 \frac{\ln^a(1-x)\ln(1+x)}{x}dx=(-1)^a a! \sum_{n=1}^\infty\frac{H_n^{(a+1)}}{n2^n}$
Nice little generalization:
$$\int_0^1 \frac{\ln^a(1-x)\ln(1+x)}{x}dx=(-1)^a a! \sum_{n=1}^\infty\frac{H_n^{(a+1)}}{n2^n},\quad a=0,1,2,...$$
The point of this post is to save us some calculations ...
- 25.8k
20
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A generalization of an integral related with $\zeta(2)$
It is well-known that:
$$ \int_{0}^{+\infty}\frac{x}{e^{x}-1}\,dx = \zeta(2) = \sum_{n\geq 1}\frac{1}{n^2} \tag{1}$$
but what is known about
$$ I_2 = \int_{0}^{+\infty}\frac{x^2}{e^x-1-x}\,dx \...
- 355k
20
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Simpson's Rule for Double Integrals
Simpson's Rule for double integrals:
$$\int_a^b\int_c^df(x,y) \,dx \,dy$$
is given by
$$S_{mn}=\frac{(b-a)(d-c)}{9mn} \sum_{i,j=0,0}^{m,n} W_{i+1,j+1} f(x_i,y_j) $$
where: $$W= \begin{pmatrix}
1&...
- 4,711
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Calculate using residues $\int_0^\infty\int_0^\infty{\cos\frac{\pi}2(nx^2-\frac{y^2}n)\cos\pi xy\over\cosh\pi x\cosh\pi y}dxdy,n\in\mathbb{N}$
Q: Is it possible to calculate the integral
$$
\int\limits_0^\infty \int\limits_0^\infty\frac{\cos\frac{\pi}2
\left(nx^2-\frac{y^2}n\right)\cos \pi xy}{\cosh \pi x\cosh \pi y}dxdy,~n\in\mathbb{N}\...
- 3,933
19
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Open problems in Federer's Geometric Measure Theory
I wanted to know if the problems mentionned in this book are solved. More specifically, at some places, the author says that he doesn't know the answer, for example :"I do not know whether this ...
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18
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A difficult integral for the Chern number
The integral
$$
I(m)=\frac{1}{4\pi}\int_{-\pi}^{\pi}\mathrm{d}x\int_{-\pi}^\pi\mathrm{d}y \frac{m\cos(x)\cos(y)-\cos x-\cos y}{\left( \sin^2x+\sin^2y +(m-\cos x-\cos y)^2\right)^{3/2}}
$$
gives the ...
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Is there any $a, b \in \mathbb R, b \ne 0$ such that $\Gamma(a+bi)$ can be evaluated manually?
Is there any $a, b \in \mathbb R, b \ne 0$ such that $\Gamma(a+bi)$ can be evaluated manually? (Like $\Gamma(\frac 12)$)
If there is/are, could you show me how to calculate it?
I found that $\Gamma(...
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Other approaches to $\int_{0}^{1} \frac{K\left ( x \right ) }{\sqrt{3-x} } \text{d}x$
Let $K(x) = \int_0^{\pi/2}\frac{1}{\sqrt{1-x^2 \sin^2 \theta}}d\theta$ be the complete elliptic integral of first kind. It could be shown that
$$
\int_{0}^{1} \frac{K\left ( x \right ) }{\sqrt{3-x} } \...
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Integral involving Complete Elliptic Integral of the First Kind K(k)
I have run into an integral involving the complete elliptic integral, which can be put into the following form after changing integration variables to the modulus:
$$\int_0^{\sqrt{\frac{\alpha}{1+\...
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