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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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Prove that the regularized incomplete beta function monotone with each of its parameter

Consider the regularized incomplete beta function $I_x(a, b)$ with $x \in [0,1]$ and $a, b > 0$. I am hypothesizing that the function is monotone decreasing with respect to $a$ and monotone ...
Shiwen Yang's user avatar
2 votes
1 answer
71 views

Proof of the monotonicity of a regularized incomplete beta function

I want to prove the monotonicity of $I_r(nr, 2+(1-r)n)$ on $n$ but has no clues. The $I$ is the regularized incomplete beta function, defined as follows: $$I_r(nr, 2+(1-r)n)=\frac{\int_0^r x^{nr-1}(1-...
J H's user avatar
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1 answer
106 views

How to prove that $ \sum_{k=1}^\infty \frac{\sin kx}{k^z} = \frac{1}{\Gamma(z)} \int_0^\infty \frac{t^{z-1}e^t\sin x}{1–2e^t\cos x+e^{2t}}dt? $ [closed]

I need it to show that $\displaystyle\sum_{k=1}^\infty \frac{\sin k}{k^3} = \frac{2\pi^2-3\pi+1}{12}$
Aryan Arora's user avatar
-2 votes
0 answers
50 views

Solve a definite integral that represents the variation of a capacitance with a given angle [migrated]

I am trying to find an explicit form of the following definite integral. I have tried Mathematica but it failed to give me an answer. I know this returns something related to a trigonometric function. ...
Luan Pacheco's user avatar
0 votes
1 answer
73 views

Orthogonal space of polynomials

Let $f \colon [0,+\infty) \to \mathbb R$ be a continuous function. Assume that for any non-negative integer $n$, the function $f(t) t^n$ in integrable in $(0,+\infty)$ and $$ \int_0^{+\infty} f(t) t^n ...
henrysupercool's user avatar
0 votes
1 answer
60 views

PDF of the difference of two Beta Prime distribution

I am struggling to find the PDF of the difference of two Beta Prime distribution. Definition A random variable is said to have a Beta Prime distribution $\text{B}'(\alpha, \beta)$ with $\alpha, \beta&...
NancyBoy's user avatar
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-1 votes
0 answers
87 views

Sufficient conditions for $1/f\in L^p$

This is a very simple question that I have not found a satisfactory answer for. When is the reciprocal of a function $f:[-1,1]\to\mathbb{R}$ in $L^p([-1,1])$? In other words, when is $\int1/f^p\,dx<...
tedl445's user avatar
1 vote
2 answers
161 views

Definite integral of Gaussian divided by hyperbolic cosine

Question: is there a nice formula for $\int_{-\infty}^{+\infty} \frac{e^{-rx^2}}{e^x + e^{-x}} dx$ ? For a real parameter $r > 0$. Maybe useful: consider the bilateral Laplace transform $$J(z) = \...
Sergio A. Yuhjtman's user avatar
-2 votes
1 answer
150 views

Averaged measure in integrations

Consider \begin{align} & F(n,x)\equiv \int_0^x \cdots g (x_5)\int_0^{x_5} ~\int_0^{x_4} g (x_3)~~\int_0^{x_3} ~\int_0^{x_2} g (x_1)~~A(x_1)\,dx_1\cdots dx_n \end{align} where $g(x)$ is a measure. ...
Math2024's user avatar
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-1 votes
1 answer
197 views

Cauchy reduction formula with measure (a variation)

The Cauchy reduction formula conveniently compresses $n$ integrations of a function $F(x)$ into a single integral. Here I am interested in reducing the following "curved-space" ...
Math2024's user avatar
  • 141
3 votes
2 answers
403 views

Closed form for $ \int_{0}^{1} \dotsi \int_{0}^{1} \frac{x_1^q + \dotsb + x_n^q}{x_1^p + \dotsb + x_n^p} \, \mathrm{d}x_1 \dotsm \mathrm{d}x_n $

I asked this question on MSE, but received no answer. Recently, reading this problem, I found out that $$ \lim_{n\to \infty} \int_{0}^{1} \dotsi \int_{0}^{1} \frac{x_1^q + \dotsb + x_n^q}{x_1^p + \...
user967210's user avatar
1 vote
0 answers
105 views

Generalization of the triangle inequality to complex exponents: What is $P\left(\left| x^{a+bi} + y^{a+bi} \right| \ge \left|z^{a+bi}\right|\right)$?

Let $x \le y \le z$ be the length of the sides of a triangle whose vertices are uniformly random on the circumference of a circle. In this question, it was proved that if $a \ge 1$, then the ...
Nilotpal Kanti Sinha's user avatar
13 votes
1 answer
718 views

If $(a,b,c)$ are the sides of a triangle, then the probability $P(ax + by \ge c) = \frac{4}{\pi^2}\chi_2(x) + \frac{4}{\pi^2}\chi_2(y)$

Posting this question in MO since it is unanswered in MSE Let $(a,b,c)$ be the side of a triangle. In its most general linear form, the triangle inequality can be expressed as: Does $ax + by \ge c$ ...
Nilotpal Kanti Sinha's user avatar
1 vote
1 answer
326 views

Cauchy reduction formula with measure

The Cauchy reduction formula is $$\int_{0}^{x}{\cdots{\int_{0}^{x_{3}}{\int_{0}^{x_{2}}{F(x_1)\,dx_1\cdots dx_n}}}} =\frac{1}{(n-1)!}{\int_{0}^{x}{(x-t)^{n-1}F(t)\,dt}}$$ Consider a generalisation $$\...
Math2024's user avatar
  • 141
-2 votes
0 answers
91 views

Indefinite integral of li(x) dependant expression

I have the following $f(x)$ function: $f(x)=\frac{4 (n-1)^x n^{1-x} \operatorname{li}\left(1-\left(\frac{n-1}{n}\right)^x\right)}{2 \text{li}\left(1-\left(\frac{n-1}{n}\right)^x\right)-3}$ When I try ...
Cardstdani's user avatar
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0 answers
34 views

Integral of gradient of a function times a vector fields, null whatever the function, implies null divergence and tangential limits conditions

I'm reposting this question from math.stackexchange, as I haven't got answers so far. At the beginning of "Brenier, Y. (1987) Décomposition polaire et réarrangement monotone des champs de ...
Oersted's user avatar
  • 101
10 votes
1 answer
666 views

Find the area of the region enclosed by $\sin^p x+\sin^p y=\sin^p(x+y)$, the $x$-axis and the $y$-axis (comes from a probability question)

Consider the graph of $\sin^p x+\sin^p y=\sin^p(x+y)$, where $x$ and $y$ are acute, and $p>1$. Here are examples with, from left to right, $p=1.05,\space 1.25,\space 2,\space 4,\space 100$. Find ...
Dan's user avatar
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0 votes
0 answers
75 views

An integral over the sphere in $\mathbb{R}^d$

Let $S^{d-1}$ be the unit sphere in $\mathbb{R}^d$. Let $|x-y|$ denote the euclidean distance between to points $x$ and $y$ in $\mathbb{R}^d$. Is there a nice expression for the following (maybe ...
user111's user avatar
  • 3,854
-1 votes
0 answers
30 views

Error on spline interpolation for integration

I am interested in studying the integral $$ I \equiv \int_m^M dx \, f(x) d(x) $$ where $f(x)$ is an analitcally known function (mildly oscillating), while $d(x)$ is sampled at a finite, discrete set ...
knuth's user avatar
  • 31
-1 votes
0 answers
67 views

Integrating $\int_{-\infty}^{\infty}\exp(-x^2/(1-ix))(1-ix)^{-1/2}{\rm d}x$

I want to calculate the integral $$ I=\int_{-\infty}^{\infty}\exp\left(\frac{-x^2}{1-ix}\right)(1-ix)^{-1/2}\ {\rm d}x. $$ I got two ways to deal this integral. The first, noting that $$ \int_{-\infty}...
Guoqing's user avatar
  • 429
3 votes
0 answers
100 views

Maximal-type inequality for a Borel probability measure supported on a subset of $L^2(\mathbb{R}^d)$

Let $\mu$ be a Borel probability measure on $L^2(\mathbb{R}^d)$ for $d\ge 1$ which is moreover supported on the unit sphere $$S=\{\phi\in L^2(\mathbb{R}^d): \| \phi\|_{ L^2(\mathbb{R}^d)}=1\}.$$ Let ...
Diffusion's user avatar
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24 votes
5 answers
2k views

Find the area of the region enclosed by $\frac{\sin x}{\sin y}=\frac{\sin x+\sin y}{\sin(x+y)}$ and the $x$-axis (comes from a probability question)

This question resisted attacks at MSE, so I am posting it here. Here is the graph of $\dfrac{\sin x}{\sin y}=\dfrac{\sin x+\sin y}{\sin(x+y)}$. Find the area of the region enclosed by the curve and ...
Dan's user avatar
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0 votes
0 answers
78 views

Possible closed form of the integral contains delta function?

I want to evaluate an $N$-dimensional integral defined by $$ P(s,k)=\int_{-\infty}^{\infty}{\rm d}x_1\dotsi\int_{-\infty}^{\infty}{\rm d}x_N\ \delta\left(\Biggl|\frac{\sum_{m\ne n}^N\exp(i(m+n)k)(x_m-...
Guoqing's user avatar
  • 429
0 votes
0 answers
129 views

Help me find the antiderivative of $W(W(x))$ where $W$ denotes the Lambert W Function

Let $W$ denote the Lambert W Function. I must know the antiderivative of $W^2 = W(W(x))$. I'm already convinced this function is not elementary. This does nothing to settle up my curiosity, as I ...
Alma Arjuna's user avatar
3 votes
1 answer
231 views

Expectation on a Polish space

I was wondering, if given a Polish space $X$, and given some probability measure $p$ on $X$, can the expectation of an $X$-valued function be taken? In particular, would the integral $\int_X x dp$ ...
uniform_on_compacts's user avatar
1 vote
0 answers
63 views

Discretization of oscillating integral

Suppose I am interested in computing $$ I \equiv \int_0^B dx \, g(x) f(x) $$ where $B$ is a known upper bound for the integral, $g(x)$ is a known oscillating function and $f(x)$ is a smooth function ...
knuth's user avatar
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0 votes
1 answer
95 views

Numerical integration with integrable singularity

Suppose I have a numerical estimation of discrete samples of a smooth function $C(t)$ at $t = a, \dots, T = Na$ and I want to (numerically) compute the integral of $f(t) = \frac{C(t)}{\sqrt{t}}$. In ...
knuth's user avatar
  • 31
4 votes
1 answer
253 views

Approximating a finite sum with an integral

Consider the following sum (with $a$ being a real number and $N$ an even integer) $$S(a, N) = \sum_{m=1}^{N/2} \frac{4}{N+1} \sin^2\left( \frac{2\pi m}{N+1} \right)\sin \left( 2 a \cos \left( \frac{2\...
Nigel1's user avatar
  • 285
2 votes
1 answer
282 views

A strange functional inequality

Let $f,g \in C([-2,2],\mathbb R_+^*)$ even and concave real functions. Is it true that $$ \int_0^1 f\big(\cos(x^{-1})+\sin(x^{-1})\big) \cdot g\big(\cos(x^{-1})-\sin(x^{-1})\big) \mathrm{d}x\\ \leq f(...
Dattier's user avatar
  • 3,767
0 votes
0 answers
69 views

Schauder basis of L^p on general domains

Let $\Omega$ be a non-empty regular compact subset of $\mathbb{R}^n$; i.e. the closure of its interior is itself; for some $n\in \mathbb{Z}^+$. Let $1\le p<\infty$. When does the space $L^p(\...
ABIM's user avatar
  • 4,775
1 vote
1 answer
103 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 ...
Daeree's user avatar
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6 votes
1 answer
250 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 ...
D.R.'s user avatar
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1 vote
0 answers
130 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 ...
lrnv's user avatar
  • 686
1 vote
0 answers
63 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 ...
GigaByte123's user avatar
1 vote
1 answer
74 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 ...
Drew Brady's user avatar
0 votes
1 answer
76 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 ...
Mario Vasilija's user avatar
5 votes
0 answers
396 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 ...
Jorge Zuniga's user avatar
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3 votes
1 answer
215 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 ...
Faoler's user avatar
  • 431
0 votes
0 answers
109 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. ...
CrSb0001's user avatar
  • 145
1 vote
0 answers
90 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)\...
Martin.s's user avatar
  • 172
4 votes
2 answers
671 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....
Martin.s's user avatar
  • 172
7 votes
1 answer
334 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^{\...
Peter Johnson's user avatar
2 votes
0 answers
156 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 $...
Pavel Gubkin's user avatar
5 votes
1 answer
341 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 ...
Martin.s's user avatar
  • 172
1 vote
0 answers
70 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 $$ ...
Pavel Gubkin's user avatar
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 ...
Ivan's user avatar
  • 689
1 vote
0 answers
116 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 ...
poeaqnwgo's user avatar
7 votes
1 answer
467 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}^{\...
Faoler's user avatar
  • 431
1 vote
1 answer
74 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 ...
Didier Felbacq's user avatar
2 votes
0 answers
90 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 ...
freeRmodule's user avatar
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