Questions tagged [integration]

Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of the integrals being considered. This tag often goes along with the (calculus) tag.

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175
votes
0answers
9k views

Is this really a categorical approach to integration?

Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A ...
51
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0answers
1k views

Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ We can use this to evaluate integrals. For example, consider $f(x)=...
31
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0answers
3k views

A difficult integral

For $\gamma>0,\delta>0$, How do I evaluate this integral? $$ I=\int_0^H\frac{e^{i t x} \log\left(\frac{H}{H-x}\right) ^{\frac{1}{\gamma }-1} \left(\left(\frac{k}{H \log \left(\frac{H}{H-x}\right)...
28
votes
0answers
972 views

Proving that $\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$

Prove without evaluating the integrals that:$$2\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx=\int_\frac{\pi}{2}^\pi\frac{x\ln(1-\sin x)}{\sin x}dx\label{*}\tag{*}$$ Or equivalently: $$\boxed{\...
22
votes
0answers
457 views

Geometric representation of Euler-Maclaurin Summation Formula

In Tom Apostol's expository article (here's a free link), upon seeing the figure below (or this from the Wolfram project) I was expecting more diagrams to come to continue the error decomposition of ...
21
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0answers
708 views

Evaluate $\int_{\mathbb R}\frac{x}{\sin^2(\sqrt{x})\sinh^2\left(2\sqrt{2x}\right)+\pi^2\cos^2(\sqrt{x})\cosh^2\left(2\sqrt{2x}\right)}\mathrm dx$

I encountered an astonishing integral (numerically verified): $$\int_{\mathbb R} \frac{x\ \mathrm dx}{\sin ^2(\sqrt{x}) \sinh ^2(2 \sqrt{2 x})+\pi ^2 \cos ^2(\sqrt{x}) \cosh ^2(2 \sqrt{2 x})}\\ \...
20
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0answers
976 views

Evaluating $\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}\,\mathrm{d}x$

Evaluate $$\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}\,\mathrm{d}x.$$ I tried using by parts and complex numbers along with series expansion but I was unable to find the answer. Please Help!
17
votes
0answers
568 views

How to prove $\int_0^1x\ln^2(1+x)\ln(\frac{x^2}{1+x})\frac{dx}{1+x^2}$

How to prove$$\int_0^1x\ln^2(1+x)\ln\left(\frac{x^2}{1+x}\right)\frac{dx}{1+x^2}=-\frac{7}{32}\cdot\zeta{(3)}\ln2+\frac{3\pi^2}{128}\cdot\ln^22-\frac{1}{64}\cdot\ln^42-\frac{13\pi^4}{46080}$$ The ...
16
votes
0answers
385 views

$\lim_{n\to\infty} \underbrace{\int_{0}^{1}\cdots \int_{0}^{1}}_{n}\frac{x_1^{505}+\cdots +x_n^{505}}{x_1^{2020}+\cdots +x_n^{2020}}dx_1\cdots dx_n$

Evaluate this multiple integral inside a limit: $$\lim_{n\to\infty} \underbrace{\int_{0}^{1}\cdots \int_{0}^{1}}_{n}\frac{ \sum _{k=1}^{n}x_k^{505}}{\sum_{k=1}^{n}x_k^{2020}} \mathrm d x_1\cdots \...
16
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0answers
506 views

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 ...
16
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0answers
321 views

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{\...
15
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0answers
403 views

Are these continued fractions of integrals known?

Motivated by this paper on polynomial continued fractions (Bowman, 2000), I thought about various extensions to current definitions of these fractions. What if we defined a continued fraction such ...
15
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0answers
1k views

Calculate using residues $\int_0^\infty\int_0^\infty{\cos\frac{\pi}2\Big(nx^2-\frac{y^2}n\Big)\cos\pi xy\over\cosh\pi x\cosh\pi y}dxdy,n\in\mathbb{N}$

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}\...
15
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0answers
7k views

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&...
15
votes
1answer
787 views

Integral involving square root of sine and cosine

Is there any closed formula for $$ \int_{0}^{\pi/2} \dfrac{e^{-x}\sqrt{\cos x}\ dx}{\sqrt{\cos x} + \sqrt{\sin x}} $$ I know $$ \int_{0}^{\pi/2} \dfrac{\sqrt{\cos x}\ dx}{\sqrt{\cos x} + \sqrt{\sin x}...
14
votes
0answers
704 views

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 ...
14
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0answers
751 views

Nontrivial trivial integrals

Consider two propositions in geometry: Circumscribe a right circular cylinder about a sphere. The surface area of the cylinder between any two planes orthogonal to the cylinder's axis equals the ...
14
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0answers
966 views

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+\...
13
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0answers
698 views

Omega Constant Integral

Whilst reading this Math SE post, I saw that the OP mentioned the integral $$\int_0^\infty \frac{1+2\cos x+x\sin x}{1+2x\sin x +x^2}dx=\frac{\pi}{1+\Omega}$$ where $\Omega$ is the unique solution to ...
13
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0answers
424 views

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 \...
13
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0answers
358 views

Integration of a $k$-form over chains

In Spivak's Calculus on Manifolds, he defines the integral of a $k$-form over a $k$-chain, and proves a version of Stokes' theorem for this situation, before moving on to discuss the integral of a ...
13
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0answers
430 views

An integration to first order

I am having some trouble evaluating an integral -- involving taking an approximation. It would be great if someone could help me. I wish to evaluate $$\int_0^\pi {\cos\theta\cos \left[\omega t-{\...
13
votes
0answers
577 views

Finding a proper solution of a given functional

It's my first post here, but I worked very hard to find solution and I failed. Hereinafter, I skip physical background and directly proceed to my mathematical problem. No matter how, you know the ...
12
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0answers
183 views

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(...
12
votes
0answers
185 views

Evaluate the following integral involving $\sin \pi x$

Let $F: \Bbb{R} \to \Bbb{R}$ be defined by $$F(s)=\begin{cases}1, & \text{if }s\ge \dfrac12 \\[0.2cm]0, & \text{if }s< \dfrac12 \end{cases}$$ I need to evaluate $$\int^{1}_{0} F(\sin \pi ...
11
votes
1answer
264 views

Proof of inversion formula in Probability

Theorem : Let $X$ be a real random variable such that $\phi_X \in L^1$ i.e $\int_{\mathbb{R}} \vert \phi_X(t) \vert < \infty$, then $X$ has density $f_X(x) \in C_b(\mathbb{R})$ given by $$f_X(x) = \...
11
votes
0answers
267 views

Tricky surface integral of vector field

We have the embedded surface $S= \{(x,y,z)\in \mathbb{R}: z = e^{1-(x^2 + y^2)^2}, z>1\}$ and the vector field $\mathbf F:\mathbb{R}^3\to \mathbb{R}^3; (x,y,z)\mapsto (x e^{y^2}, 2ye^{x^2}, 5-3z) $....
11
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0answers
712 views

Theorem 6.12 (a) in Baby Rudin: $\int_a^b \left( f_1 + f_2 \right) d \alpha=\int_a^b f_1 d \alpha + \int_a^b f_2 d \alpha$

Here is part (a) of Theorem 6.12 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: If $f_1 \in \mathscr{R}(\alpha)$ and $f_2 \in \mathscr{R}(\alpha)$, then $$f_1 + ...
11
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0answers
410 views

A very useful lemma for Henstock-Stieltjes integration

I'd like to see a proof (or hints and outlines) for the following lemma, which is very useful to prove some interesting properties, including an Integration by Parts theorem for Henstock-Stieltjes ...
11
votes
0answers
649 views

Egorov's theorem for this Lebesgue integral

I want to prove Egorov's theorem using this Lebesgue integral defined by the upper integral $$\int^*f:=\left\{\int h ; h \ge f \text{ and h upper-continuous }\right\}$$ $$\int_*f:=\left\{\int h ; h \...
11
votes
0answers
677 views

Change of variables for stochastic integral

Let $H$ be a previsible locally bounded process, and let $X$ be a continuous local martingale. If $T$ is a stopping time and $X^T=(X_{t+T}-X_{T},t\geq 0) $ then $$\int_T^{t+T}H_s.dX_s=\int_0^tH_{...
11
votes
0answers
477 views

Is this question solvable? $2$ non-linear equations and the proof that the solution is unique

As mentioned in the title I want to show the uniqueness of the solution to $2$ non-linear equations. However, it seems that I can not solve this question with my current mathematical knowledge. More ...
11
votes
1answer
345 views

Evaluating $ \int_0^{\infty}\frac{v}{\sqrt{v + c}}e^{-\frac{y^2}{2(v + c)} - \frac{(u-v)^2}{u^2v}}dv$

While working on mixture (variance) of normal distribution and keep running into these two integrals $$ \int_0^{\infty}\dfrac{v}{\sqrt{v + c}}e^{-\dfrac{y^2}{2(v + c)} - \dfrac{(u-v)^2}{u^2v}}dv,$$ $...
11
votes
0answers
1k views

How to compute this integral of Bessel functions?

I have $\alpha_\max$ a real number between $0$ and $\frac\pi2$. Furthermore $\zeta$ and $\xi$ are positive real numbers. Now I would like compute the integral $$\int_0^{\alpha_\max} \mathrm{e}^{i \...
11
votes
0answers
1k views

Cauchy-Formula for Repeated Lebesgue-Integration

Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given. Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...
10
votes
0answers
249 views

Can we further refine $\int_{0}^{1}x^{x^x} \ dx=\frac 1 2+\sum_{n=1}^{\infty}(-1)^n\sum_{k=1}^{\infty}\frac {(n-k)^k}{(k+1)^{n+1}} \binom {n}{k}$

Question Can we further refine the integral $$\int_{0}^{1}x^{x^x}\ dx=\frac 1 2+\sum_{n=1}^{\infty}(-1)^n\sum_{k=1}^{n}\frac {(n-k)^k}{(k+1)^{n+1}} \binom {n}{k}$$ ? To compute the result, first ...
10
votes
0answers
80 views

Recursive integrals a la $\int_{\int_y^xg(t)dt}^{\int_x^yg(t)dt}g(t)dt$

Inspired by this question I was wondering whether "recursive" integrals have been studied or if they appear anywhere in applications. What I mean is the following: Let $I(x, y) = \int_x^y g(t) dt$ and ...
10
votes
1answer
295 views

$I = \int_0^k z^{m_1 - 1} \ln(1 + z) \left(\frac{m_1 z}{a} + \frac{m_2}{b} \right)^{-(m_1 + m_2)} \mathrm dz.$

Question: How to find the closed-form solution for the given integral? $$I = \int_0^k z^{m_1 - 1} \ln(1 + z) \left(\dfrac{m_1 z}{a} + \dfrac{m_2}{b} \right)^{-(m_1 + m_2)} \mathrm dz,$$ where $k, a, b,...
10
votes
0answers
180 views

Integral of $\int_0^{\infty} \ln\left|\frac{x+A}{x+B}\right|\frac{x}{e^{C x}\pm 1}dx$

so I have this integral to try and evaluate $$(*)=\int_0^{\infty} ln\left|\frac{x+A}{x+B}\right|\frac{x}{e^{C x}\pm1}dx$$ So far, I have managed to evaluate a very similar integral $$\int_0^{\infty}...
10
votes
0answers
131 views

How weird can the boundary be so that the fundamental theorems of vector calculus hold?

Let $\Omega$ be a connected open set in $\Bbb R^n$. Suppose that I want theorems in multivariables calculus like divergence theorem or its relative like Green's identities or even Stoke's theorem to ...
10
votes
0answers
423 views

Methodologies to Evaluate $\lim_{L\to \infty}\int_0^\infty \frac{\sin(Lx)}{x}\cos(x^3/3)\,dx$

In This Answer, I wrote "It is straightforward to show that $\displaystyle \lim_{L\to \infty}\int_0^\infty \frac{\sin(Lx)}{x}\,\cos(x^3/3)\,dx=\frac\pi2$." For completeness, I've included the "...
10
votes
1answer
177 views

Double integral - transformation

I'm trying to calculate $$\iint_{\Omega } e^{(x+y^2)^{3/2}} \,\mathrm{d}A,$$ where $$\Omega =\{x,y>0 : x+y\leq 2\}. $$ Not sure where to go with it. I need to find a transformation and then ...
10
votes
1answer
384 views

An extremely mysterious integral: $\int_0^1 \frac{k \tan^{-1}(t)}{k^2 + t^2}\mathrm d t$

$$f(n) = \int_0^1 \frac{n \tan^{-1}(t)}{n^2 + t^2}\mathrm d t \tag{n > 2}$$ Introduction: This is one of the most beautiful and mysterious integrals I've every encountered. It's very simple, but ...
10
votes
0answers
424 views

Juantheron-like integral

While seeing this post, the following integral is just struck me \begin{equation} \int_0^\infty \frac{dx}{(1+x^2)(1+\tan x)}\tag1 \end{equation} I have tried like what user @OlivierOloa did in his ...
10
votes
0answers
1k views

Integral of rational function over $\mathbb{H}^4$

Suppose I have a rational function of $8$ coordinates $a,b,c,d,e,f,g,h$ that I want to integrate over $\mathbb{H}^4$: ...
10
votes
0answers
1k views

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: $$p(y)=\int_0^{\frac{1}{\sqrt{2 \pi }}} \frac{\sqrt{\frac{2}{\...
10
votes
0answers
827 views

Nasty Integral - Closed form solution?

Any suggestions on how to integrate this beast?: $$\int_0^{\omega_t}\int_{\omega_t}^f\sin^2\left(\frac{\omega_{12}}{2}\right)\sin^2\left(\frac{\omega_{23}}{2}\right)d\omega_{23}d\omega_{12}$$ where: ...
9
votes
0answers
247 views

How do I solve this integral equation?

Case 1: We will look at an easier problem first. Let $|\alpha|, |\beta| \leq \alpha_c, \alpha_c \leq \pi$. I want to solve for $\rho(\beta)$ in the following equation, where $P$ denotes the principal ...
9
votes
0answers
270 views

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 ...
9
votes
0answers
225 views

About the product of two Elliptic integrals

Let $z,x\in\left(0,1\right)$. It is possible to prove that $$\int_{0}^{1}\int_{0}^{1}\frac{1}{\sqrt{hy\left(1-h\right)\left(1-y\right)}}\frac{dydh}{\sqrt{\left(1+zhy\right)^{2}-4xzhy}}=\frac{4}{\pi^{2}...

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