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 ...

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101
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0answers
3k views

Generalizing $\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{\operatorname dx}{x^{2}+1} = \frac{5\pi^{2}}{96}$

The following integral \begin{align*} \int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{dx}{x^{2}+1} = \frac{5\pi^{2}}{96} \tag{1} \end{align*} is called the Ahmed's integral ...
32
votes
0answers
1k 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 Categorical ...
20
votes
0answers
426 views

Is there a closed form for this sum?

While generalizing the previous result, I conjectured that the series expansion of \begin{align*} \int_{0}^{\frac{\pi}{2}} \arctan \left( \frac{2x \sin\theta}{1-x^{2}} \right) \arctan \left( \frac{2y ...
16
votes
0answers
308 views

We have sums, series and integrals. What's next?

We know how to sum or average a finite number of terms: sums. We know how to sum a countable infinite number ${\beth_0}$ of terms: series. We know how to sum ${\beth_1}$ terms: integrals. How to ...
15
votes
0answers
185 views

Integrating a matrix function involving a determinant and exponential trace

I am trying to find the normalizing constant for a probability distribution and ran into a difficult integral. When $X$ is an $p \times k$ matrix, $a>0,$ and $g>0,$ how can I compute $$\int \...
14
votes
0answers
305 views

How to find $\lim_{n \to \infty} \int_0^1 \cdots \int_0^1 \sqrt{x_1+\sqrt{x_2+\sqrt{\dots+\sqrt{x_n}}}}dx_1 dx_2\dots dx_n$

Here I mean the limit of the following sequence: $$p_1=\int_0^1 \sqrt{x} ~dx=\frac{2}{3}$$ $$p_2=\int_0^1 \int_0^1 \sqrt{x+\sqrt{y}} ~dxdy=\frac{8}{35}(4 \sqrt{2}-1) = 1.06442\dots$$ $$p_3=\int_0^...
14
votes
0answers
299 views

A closed form of $\sum_{k=1}^\infty \psi^{(1)} (k+a)\psi^{(1)} (k+b)$?

The following result $$ \sum_{k=1}^\infty\left(\psi^{(1)} (k)\right)^2 = 3\zeta(3) $$ where $\psi^{(1)}$ is the polygamma function makes me think there is a nice sum for the series $$ \sum_{k=1}^\...
14
votes
0answers
569 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
votes
0answers
570 views

Analytic form of: $ \int \frac{\bigl[\cos^{-1}(x)\sqrt{1-x^2}\bigr]^{-1}}{\ln\bigl( 1+\sin(2x\sqrt{1-x^2})/\pi\bigr)} dx $

Background: On my quest to solve difficult integrals, I chanced upon this site: http://www.durofy.com/5-most-beautiful-questions-from-integral-calculus/ Good problems for me, (novice), although I ...
13
votes
0answers
451 views

Computing the volume of a region on the unit $n$-sphere

I would like to compute the surface volume of a region on the unit $n-1$-sphere: $$\sum_{i=1}^n x_i^2 = 1,$$ bounded by an ellipsoid $$\sum_{i=1}^n a_ix_i^2 \leq a_2,$$ where $1=a_1 < a_2 <...
12
votes
0answers
257 views

Nested root integral $\int_0^1 \frac{dx}{\sqrt{x+\sqrt{x+\sqrt{x}}}}$

The bigger goal is to find the antiderivative: $$\int \frac{dx}{\sqrt{x+\sqrt{x+\sqrt{x}}}}~~~~~(*)$$ But I can settle for the definite integral in $(0,1)$. Motivation: $$\int \frac{dx}{\sqrt{x+\...
12
votes
0answers
122 views

References about Iterating integration, $\int_{a_0}^{\int_{a_1}^\vdots I_1dx}I_0\,dx$

Are there any references that discuss Iterating integration in general, $\int_{a_0}^{\int_{a_1}^\vdots I_1dx}I_0\,dx$, conditions in which they converge, some special values, some special tricks to ...
12
votes
0answers
129 views

Prove: $\frac{p}{2\pi}\int_{-\infty}^{+\infty}\frac{\sin xt}{t\cdot \sin\frac12pt}\sin([\frac xp]+\frac12) pt \mathrm dt=\cdots$

Suppose $p>0$, define that $$ g(x)=\begin{cases} p\left\lfloor\frac xp\right\rfloor+\frac p2,x\geqslant0\\\\-g(-x), x<0\end{cases}$$ Prove for all $x$, $$\displaystyle\frac{p}{2\pi}\int_{-\...
12
votes
0answers
195 views

Dilogarithm identity containing the tribonacci constant

The motivation of this question is the brilliant conjecture by @Tito Piezas III. In $(4)$ of his question the equation seems to be true for all $n > 1$ real numbers. The case $n=2$ leads us to a ...
12
votes
0answers
219 views

Why are the fundamental theorems of calculus usually associated to the Riemann Integral?

I am writing a "textbook" on Analysis, and I've reached the time I must talk about integrals. I prefer to approach directly the Lebesgue Integral theory. This question is not about the status of this ...
11
votes
0answers
298 views

Closed form of a difficult definite integral

I'm looking for a closed-form expression for the value of this integral: $$I=\int_0^\pi \frac{\sin(x)}{\sqrt{x^3+x+1}} dx$$ The graph of the integrand looks like this: $\hskip 2.4 in$ Numerically, ...
10
votes
0answers
85 views

A version of Ampère's law

The most common proof that I have found of the fact that Ampère's law is entailed by the Biot-Savart law uses the fact that, if $\boldsymbol{J}:\mathbb{R}^3\to\mathbb{R}^3$, $\boldsymbol{J}\in C_c^2(\...
10
votes
0answers
791 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
120 views

Proof of the relation $\int^1_0 \frac{\log^n x}{1-x}dx=(-1)^n~ n!~ \zeta(n+1)$

I had the thought that by introducing some parameters into simple integrals and taking derivatives we can get exact values for infinitely many 'complicated' integrals. $$\int_0^1 x^a dx = \frac{1}{a+...
10
votes
0answers
293 views

Calculate $\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx$

Prove that: $$ I=\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx=\frac{7}{2}\zeta(3){\log^22}-\frac{\pi^2}{6}{\log^32}-\frac{\pi^2}{2}\zeta(3)+{6}\zeta(5)-\frac{\pi^4}{48}\ln2 $$ Using ...
10
votes
0answers
342 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 ...
10
votes
0answers
261 views

A difficult integral $\int_0^\infty \mathrm{d}t\frac{1}{t}\frac{1}{t-s-\mathrm{i}\epsilon}\frac{1}{X}\ln\frac{1-X}{1+X} $

Can anyone give any hints on how to rewrite this in terms of dilogarithms? $$ \int_{0}^{\infty}{{\rm d}t \over t}\,{1 \over t - s - {\rm i}\epsilon}\, {1 \over \,\sqrt{\, 1 - a/t\,}\,}\, \ln\left(1 - ...
10
votes
0answers
220 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-{\...
10
votes
0answers
470 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 ...
10
votes
0answers
804 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 \...
10
votes
0answers
925 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 ...
9
votes
0answers
132 views

Using multiple integrals for tough single integrals

I'm just getting started on double integrals, and I recently saw the super cool way to use double integrals to arrive at $$\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$$ So, I am wondering if ...
9
votes
0answers
116 views

Can you integrate on a scheme?

As the question suggests, can you integrate on a scheme? How? I don't even know if this is even a well-posed question...
9
votes
0answers
252 views

Interesting Integral with Parameters

I would like to compute the following integral: $$\int\frac{d^{2}\overrightarrow{q}}{2\pi}\int\frac{d^{2}\overrightarrow{k}}{2\pi}e^{i\overrightarrow{q}\cdot\overrightarrow{r}}\left(e^{i\...
9
votes
0answers
77 views

Decay of amplitude integral

Consider the function $$ f(\vec{x}) = \int_{\Bbb R^3} {\frac{ e^{-i\,\vec{x}\cdot\vec{k}}}{\sqrt{\vec{k}^2 + m^2}}} d^3 k $$ from Zee's Quantum Field Theory in a Nutshell. He argues like this: “...
9
votes
0answers
176 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)=...
9
votes
0answers
258 views

Is each “elementary + finite functions” function “elementary + finite functions”-integrable?

It is known that there exists elementary functions which are not elementary integrable, i. e. there exists no elementary anti derivative. Example: $f(x) = e^{-x^2}$. Let $A$ be the set of elementary ...
9
votes
0answers
133 views

how to solve this indefinite integral?

Could anyone help me how to solve this indefinite integral? $$\int{dx\over \sqrt{\sin^3 x+\sin (x+\alpha)}}$$
9
votes
0answers
226 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_{...
9
votes
0answers
353 views

Is this question solvable? $2$ non-linear equations and the proof that the solution is unique (with asymmetric bounty option)

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 ...
8
votes
0answers
67 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 ...
8
votes
0answers
171 views

How to evaluate the integral $\int_0^\infty \frac{x^{a-1}}{1+bx^a} e^{-x} dx$

How to evaluate this integral? \begin{equation} \int_0^\infty \frac{x^{a-1}}{1+bx^a} e^{-x} dx \end{equation} I think it will use a gamma function or a exponential integral. I really need an ...
8
votes
0answers
410 views

${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$

I have found the following new result connecting two rational log-cosine integrals. Proposition. \begin{align} \displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos ...
8
votes
0answers
386 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 \...
7
votes
0answers
135 views

Stokes' Theorem and Vector Fields with Jump Discontinuities

What are the continuity requirements on a vector field $\boldsymbol{A}$ such that Stokes' theorem, $$ \iint_S\nabla\times\left[(\boldsymbol{\hat{x}}\cdot\nabla\phi)\boldsymbol{A}\right]\cdot d\...
7
votes
0answers
99 views

Infinite Integration in Limits of Integration

Given the following: $$ u_0 = \int \limits_{ 0 } ^{ 1 } x \, dx , \:\:\: u_1 = \int \limits^{ \int \limits_{ 1/2 } ^{ 1 } x \, dx } _{ \int \limits_{ 0 } ^{ 1/2 } x \, dx } x \,dx , \:\:\: u_2 = \int \...
7
votes
0answers
94 views

Can you add new functions to the set of elementary functions such that every function has an anti-derivative?

Its fairly well known that not every elementary function has an elementary anti-derivative. The common examples of this are $\exp(-x^2)$ and $\sin(x)/x$. The general workaround to this problem is to ...
7
votes
0answers
192 views

Help with the integral $\int_{0}^{\infty}\frac{\log(1\pm ix)^{2}}{\left(\frac{t}{2}\log(1 \pm ix) \right )^{2}-\pi ^{2}n^{2}}e^{-2\pi mx}dx$

Referring to a previous question, i want help with the integral : $$\int_{0}^{\infty}\frac{\log(1\pm ix)^{2}}{\left(\frac{t}{2}\log(1 \pm ix) \right )^{2}-\pi ^{2}n^{2}}e^{-2\pi mx}dx$$ Where $n,m$ ...
7
votes
0answers
91 views

Fredholm integral?

If one exists, find a continuous, bounded function $f: \mathbb{R} \to \mathbb{R}$ which is not identically zero and which satisfies$$0 = \int_0^\infty K(t, s)f(s)\,ds$$for all $t \in \mathbb{R}$, ...
7
votes
0answers
41 views

Equivalence class of definite integrals

Let's assume we have a smooth function $f(x):[a,b]\to \mathbb{R}$ so that the integral $$\int_a^b f(x) dx$$ is finite. By performing various changes of variables, we can derive a large (infinite?) ...
7
votes
0answers
2k views

A difficult integral

For $\gamma>0,\delta>0$, trying to 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}\...
7
votes
0answers
101 views

Can these two indefinite integrals be evaluated in closed form?

I'm wondering whether any of these two indefinite integrals $$\int \frac{1}{\sqrt{1+\alpha \sinh(x)^{-4/3}}}dx$$ $$\int \frac{\sinh(x)^{-4/3}}{\sqrt{1+\alpha\sinh(x)^{-4/3}}}dx$$ can be evaluated in ...
7
votes
0answers
451 views

Integration of product of functions(Special form)

Sir, I have been doing a proof related to one research topic. But after a long effort, I got ended up in a messy integration equation. Could you give me some suggestions to solve this equations? (Any ...
7
votes
0answers
244 views

Second derivative of Hypergeometric function

I'm looking for the following second derivative $$ \kappa_2 := \left . \frac{d^2}{d\lambda^2} \ln \left({_2F_1}\!\left(\tfrac{1}{2},\,- \lambda;\,1;\,\alpha\right)\right) \right \vert_{\lambda = 0}...
7
votes
0answers
80 views

Separable non-linear ODE (with radicals)

I am trying to solve the equation $$ \frac{dy}{dt}=\sqrt{(\gamma-1+\frac{2\alpha\beta}{2\alpha-1})e^{-2\alpha y}-\frac{2\alpha\beta}{2\alpha-1}e^{-y}+1} $$ [1] $y(0) = 0$; $t_{0}=0$; $\alpha$, $\...