Questions tagged [integration]

Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...

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Calculate special integrals

Prove:$$\int_0^1\frac{1-\cos x}{x} \, dx-\int_1^{+\infty}\frac{\cos x}{x} \, dx=\gamma, \\ \gamma=\lim_{n\to\infty}(1+1/2+\cdots+1/n-\ln n)$$ I try to use the Taylor expansion of $1-\cos x$ and ...
0 votes
1 answer
70 views

An inequality for the Darboux integral

As we know, if $f(x)$ are Riemann integrable, we have \begin{gather} \left|\int_a^b f(x)~\mathrm{d}x\right|\leq \int_a^b |f(x)|~\mathrm{d}x. \end{gather} So, for Darboux integrals, such as the upper ...
1 vote
1 answer
188 views

Prove the limit of the integral

Suppose: $f:[0,\frac{\pi}{2}]\to \mathbb{R}$, $f(x)\in C^2[0,\frac{\pi}{2}]$, $\gamma=\lim_{n\to \infty}(1+1/2+\dotsb+1/n-\ln{n})$. Prove:$$ \lim_{s\to +\infty}\left(\int_{0}^{\frac{\pi}{2}} \frac{\...
6 votes
1 answer
198 views

Densities, pseudoforms, absolute differential forms and measures, differential forms, etc

Apologies if this question is too basic, but I figured I first heard of most of these concepts on MO, so perhaps I can ask here. Gelfand’s definition, copied from AlvarezPaiva [My edit, could be ...
-3 votes
0 answers
120 views

Extending the proof of Maschke's Theorem from finite groups to algebras

In the theory of representations of a finite group there is Maschke's Theorem that any finite-dimensional representation of a finite group $G$ can be decomposed into a direct sum of irreducible ...
-1 votes
0 answers
80 views

Is it possible to evaluate $ \int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{1+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}? $ [closed]

People suggested me to upload this question on math overflow and many people gave the numerical results by desmos and wolfram alpha Motivation for the problem here: Is it possible to evaluate $$ \...
8 votes
2 answers
2k views

Solving 'impossible' integrals with a new (?) trick

The following identities have been suggested based on formulas in a previous question of mine. If complex $\theta_1=\cos^{-1}(p)$ and $\theta_2=\sec^{-1}(p)$, where $p\in(-1, 0) \cup (1, \infty)$, ...
1 vote
0 answers
129 views

Can this integral be solved analytically

I have an integral of the form $$\int_{t_1}^{t_2} \frac{\sum_{i=1}^n a_i e^{b_i t}}{\sum_{i=1}^n c_i e^{d_i t}} dt$$ Where $a_i,b_i,c_i,d_i$ are $4n$ real constants, and $t_1,t_2$ are positives. Is ...
1 vote
0 answers
42 views

Derivative with respect to initial condition for the solution of an SDE

Suppose we have an SDE (assuming the Lipschitz continuous conditions required for the existence of the solution): \begin{align} dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t \end{align} and define its ...
5 votes
0 answers
363 views

How to prove these identities for $\log(2)$ based on $_3F_2$ integrals?

In this MO post I have placed 4 Ramanujan-type hypergeometric series found using the LLL algorithm for fast computing of some logarithms. I could prove 3 of them by means of classical methods based on ...
4 votes
2 answers
625 views

Conjectured closed form of $\int\limits_0^1 \frac{\ln y \operatorname{Li}_2 (-y)}{1-y^2} \, dy$

Let's state with $\psi^{(1)}$ the trigamma. Calculate the order: $$\mathcal{S} = \sum_{n=1}^\infty (-1)^{n-1} (\psi^{(1)}(n))^2$$ (Cornel Ioan Valean) I uploaded this question here https://math....
1 vote
1 answer
67 views

Calculating an integral involving Haar measure on orthogonal projections

Let $\sigma_{n, m}$ denote the uniform measure over matrices $U \in \mathbb{R}^{n \times m}$ satisfying $U^T U = I_m$. Let $1_k \in \mathbb{R}^k$ denote the vector with all entries equal to $1$. I am ...
0 votes
1 answer
53 views

Uniform convergence of differential quotients in $L^1$

I know that the question arose already in other contexts. However, I think this question might be different. If I have $f\in W^{1,1}$ then it is obvious that $\frac{f(x+t)-f(x)}{t}$ converges ...
1 vote
1 answer
344 views

A question about eigenvalue equation of Hankel transform

When we think about the Fourier transform in two dimensional polar coordinates, the Hankel transform is the transformation with respect to the polar diameter. Now I have a question, why is the ...
3 votes
1 answer
211 views

How to find the coefficient of $x^k$ in the expression $\prod_{p=2}^n (1+xp) $

I got this general formula for $ n\in N$ (I showed it here) $$\int_0^1 \left(\frac{x}{1-x} \ln x \right)^n dx=n \sum_{p=0}^{n-1}a(n,p+1) (-1)^{n-p} \zeta(p+2)+n! $$ where $a(n,k)$ is the coefficient ...
0 votes
0 answers
106 views

Is there a closed form of $\int\frac{x\ln x}{\arctan x}dx$?

Consider the integral$$\int\dfrac{x\ln x}{\arctan x}dx\label1\tag1$$While I have made some progress on computing the antiderivative, I would like to know whether or not there is a closed form of it. ...
1 vote
0 answers
84 views

Conjectured closed form of $\int_0^1\frac{\ln^3(1+x)\,\ln^3x}x\mathrm{d}x$

I posted this question on Math Stack Exchange, but there were no helpful comments or answers https://math.stackexchange.com/q/4874446/1298448 How to integrate $${\displaystyle \int_0^1\frac{\ln^3(1+x)\...
1 vote
0 answers
85 views

$f \in L^2(X\times Y,\mu \times K)$ for Kernel $K$, is the map $X \ni x \mapsto (f(x,\cdot),x) \in \bigsqcup_{x \in X}L^2(Y,\Sigma_Y,K_x)$ measurable?

Let $(X,\Sigma_X)$ and $(Y,\Sigma_Y)$ be two measurable spaces, let $\mu$ be a measure on $(X,\Sigma_X)$, and let $(K_x)_{x \in X}$ be a transition kernel from $(X,\Sigma_X)$ to $(Y,\Sigma_Y)$, that ...
22 votes
1 answer
2k views

A difficult integral for the Chern number

Cross post from Maths stack exchange The integral $$ I(m)=\frac{1}{4\pi}\int_{-\pi}^{\pi}\mathrm{d}x\int_{-\pi}^\pi\mathrm{d}y\phantom{,} \frac{m\cos(x)\cos(y)-\cos x-\cos y}{\left( \sin^2x+\sin^2y +...
11 votes
1 answer
502 views

A function with unexpectedly simple Legendre transformation

Let $I(x) = \frac{1}{2\pi} \int_{-2}^2 \sqrt{4-y^2}\ln|x-y|dy$. Then $I(x)$ is a concave function and \begin{equation} I(x)= \begin{cases} \frac{1}{4}x^2-\frac{1}{2}, &\text{if } |x|\leq2 \\ \...
2 votes
0 answers
133 views

Taylor coefficients of the integral of the ordered exponential

Let $A$ be a continuous $2\times 2$ matrix-valued function on $[0,1]$. Define $X_A$ as the solution of $$ X_A'(t) = A(t) X_A(t), \qquad X(0) = I. $$ In other words $X_A$ is the ordered exponential of $...
7 votes
1 answer
327 views

Exponential trigonometric integral

I want to compute the normalization constant of some probability density on SO(3). After some simplification, I arrive at the following double integral: $$ \tag{1}\label{eq:1} \int_0^{2 \pi} \int_0^{\...
65 votes
2 answers
23k views

Does there exist a complete implementation of the Risch algorithm?

Is there a generally available (commercial or not) complete implementation of the Risch algorithm for determining whether an elementary function has an elementary antiderivative? The Wikipedia article ...
5 votes
1 answer
311 views

Conjectured closed form of $\int_0^1 \frac{\text{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$

After reading some meta posts, I've decided to post this question on MathOverflow since I didn't receive any comments or answers on MSE Certainly, I apologize for any oversight. Here's a more refined ...
1 vote
0 answers
69 views

Integral inequality related to the (mixed?) moments of two functions

For $k\in \mathbb{Z}_+$ and $t\in [0,1]$ we set $$ S_{k}(t) = \{(t_1,\ldots, t_k)\colon 1\ge t\ge t_1\ge\ldots\ge t_k\ge 0\}. $$ Let $a$ and $b$ be two continuous functions on $[0,1]$. Introduce $$ ...
0 votes
1 answer
98 views

Demonstrating bound on integral

I found this question on another forum and it got me interested because of how tight the bound is: prove that $$\int_0^\infty \frac{\arctan(x)}{x^2 + 4}\,dx > \frac{\pi}{4}.$$ The difference ...
87 votes
8 answers
15k views

Why is Lebesgue integration taught using positive and negative parts of functions?

Background: When I first took measure theory/integration, I was bothered by the idea that the integral of a real-valued function w.r.t. a measure was defined first for nonnegative functions and only ...
1 vote
1 answer
586 views

Integral on level sets

Let $g_\epsilon : K \subset \mathbb{R}^d \rightarrow \mathbb{R}$ (more regularity can be assumed if necessary) be defined on a compact set (with regular boundary) $K \subset \mathbb{R}^d$, and the ...
1 vote
0 answers
112 views

Integration in polynomial time

The work of Friedman and Ko and Müller guarantee the polynomial time computability of the integrals of analytic functions inside the circle of convergence. But do algorithms have practical value? Is ...
7 votes
1 answer
453 views

How to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?

Here in my answer I got the real part for the polylogarithm function at $1+i$ for natural $n$ $$ \Re\left(\text{Li}_n(1+i)\right)=\left(\frac{-1}{4}\right)^{n+1}A_n-B_n $$ where $$ B_n=\sum_{k=0}^{\...
11 votes
1 answer
352 views

Slick proofs using the Henstock–Kurzweil integral?

I enjoyed Iosif Pinelis's slick answer to another MO problem using the Henstock–Kurzweil integral. Are there other examples of problems whose statement does not explicitly involve the Henstock–...
42 votes
6 answers
10k views

Why do I need densities in order to integrate on a non-orientable manifold?

Integration on an orientable differentiable n-manifold is defined using a partition of unity and a global nowhere vanishing n-form called volume form. If the manifold is not orientable, no such form ...
1 vote
0 answers
54 views

Characterization of an integral operator with a Bessel kernel

I am considering the following integral operator: $$K(\sigma)(\theta)=\int_0^{2\pi} \sigma(\theta') J_0(|e^{i\theta}-e^{i\theta'}|)\,d\theta',$$ where $J_0$ is the Bessel function of order $0.$ I am ...
14 votes
1 answer
1k views

The perturbation of non-Hamiltonian algebraic vector fields

In this question, we are interested in the number of limit cycles which appears in the following perturbational system: \begin{equation}\cases{ x'=y -x^{2}+\epsilon P(x,y) \\ y'=-x+\epsilon Q(x,y) } \...
0 votes
2 answers
479 views

On integral relating logarithm of absolute value of Zeta function

Sorry for such a direct question: Consider the following integral: $$I(t)=\int_{1/2}^{1} {\log|\zeta(a+it)|}da.$$ How to find the nature of $I(t)$ as $t\rightarrow\infty$?
2 votes
0 answers
88 views

Techniques for computing integrals on $G/K$

Edit: I originally tried writing a simpler question, but to make more clear what I want I wrote the general question below Let $G$ be a compact group with compact subgroup $K\subseteq G$ (everything ...
1 vote
3 answers
571 views

limit of definite integral as $N \to \infty$

I'm interested in $\theta(N):=\int_0^1 (1-x)^{N-1} e^{xN} dx$. I'd like to show that $\theta(N)\sim c/\sqrt{N}$ as $N\to\infty$ and determine $c$. Any ideas?
1 vote
1 answer
198 views

Antiderivative of Meijer G-function

In the python sympy CAS framework one strategy to compute integrals is to transform the integration kernel to a MeijerG-function, obtain the corresponding antiderivative as MeijerG-function and ...
9 votes
0 answers
1k views

How complicated can an elementary antiderivative get?

I asked this question on MSE here. I recently learned that there are many very large numbers that have been defined, such as $\operatorname{TREE}(3)$ and many others that are too big to be written ...
0 votes
0 answers
45 views

Computing the Laplace transform of an expression

I would like to find the Laplace transform of the following expression with respect to the Laplace parameter s $ \int_{z=u}^{\infty} e^{-az/c} g^{'}(\dfrac{z-u}{c}) \int_{x=0}^{\infty} \varphi(z-x)dF(...
8 votes
2 answers
805 views

A curious family of integrals that give $\pi$

I have noticed experimentally that: $$\int_0^1 \frac{\color{red}{x}}{x^4+2x^3+2x^2-2x+1} \,dx=\color{blue}{\frac{\pi}{8}},\tag{1}$$ $$\int_0^1 \frac{\color{red}{1-x^2}}{x^4+2x^3+2x^2-2x+1} \, dx=\...
7 votes
2 answers
282 views

Étendue measure of the set of lines between two Euclidean balls

Let $d>0$ and $r_1,r_2>0$ such that $r_1+r_2 < d$. Consider two (say, closed) balls $B_1,B_2$ in $\mathbb{R}^m$ having radii $r_1,r_2$ and whose centers are at distance $d$. Let $C$ be the ...
2 votes
2 answers
159 views

Integral of nested trigonometric function $\frac{\cos\left(p\cos\left(x\right)\right)}{p\cos\left(x\right) + c}$

while still working on my problem $\int J_{0}\left(\alpha x\right)J_{0}\left(\beta x\right)\cos(kx)\,\mathrm{d}x$ I came across the following definite integral \begin{equation} \int_{0}^{\pi}\frac{\...
1 vote
2 answers
161 views

Asymptotic estimation of an integral

I have an integral of the form $$ I = \int\limits^{1}_{0} \exp\left(\dfrac{vt}{(v+1)^2 + v^2} - vt\right) dv $$ and I want to prove that $I\leq c t^{-1}$ for the sufficiently large $t$, where $c$ is a ...
3 votes
2 answers
525 views

Upper bound for complex integral

I am interested in obtaining a good upper bound for the absolute value of the following integral $$ \left| \int_{0}^{\pi/3} e^{-itn} \left( 1-e^{it} \right)^{k} dt \right|, $$ when $n>k>0$ are ...
9 votes
1 answer
1k views

Integration by parts formula for the double Riemann-Stieltjes integral

In my research the following integration by parts formula for the double Riemann-Stieltjes integral $$\int\limits_{[a,b]\times[c,d]}f(x,y)\,dg(x,y)=f(b,d)g(b,d)-f(a,d)g(a,d)-f(b,c)g(b,c)+f(a,c)g(a,c)...
2 votes
2 answers
257 views

Definite integral of Bessel function of the first kind times $x^{3/2}$

I am looking for preferably a closed form (or series solution if not possible) for the following integral: $$\int_0^a x^{3/2} J_\nu (bx) dx$$ where $\nu$ is an integer. This 1D integral appears when ...
2 votes
0 answers
936 views

On a deceptively tricky calculus problem

Motivation for this question: If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dots B_n'$ also satisfies the same inequality Let $A$ be a non-constant operator acting on $C^...
3 votes
1 answer
494 views

Does the integral $\int_0^{\infty}e^{cx^2+dx}dx/(a+bx)$ have a closed form?

The integral is $$\DeclareMathOperator{\dm}{d\!} \int_0^{\infty}\frac{e^{-cx^2+dx}}{a+bx}\dm x. $$ Here I assume that $a,b,c,d$ are chosen to make this integral convergent. Rewritting the rational ...
0 votes
1 answer
123 views

Solution or approximation to $\int x^{-a} \Gamma\left( b, c x^{-d} + e \right) dx$

I'm looking for a solution or approximation to the following indefinite integral $$\int x^{-a} \Gamma\left( b, c x^{-d} + e \right) dx.$$ I've tried Mathematica, but it does not converge to a solution....

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