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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94
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2k 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 ...
18
votes
0answers
404 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 ...
15
votes
0answers
286 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 ...
14
votes
0answers
169 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
288 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 $$ ...
14
votes
0answers
565 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: ...
13
votes
0answers
447 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
115 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
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122 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 $x$, ...
12
votes
0answers
191 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
211 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 ...
12
votes
0answers
560 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 ...
11
votes
0answers
286 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
79 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 ...
10
votes
0answers
785 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
222 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 ...
10
votes
0answers
113 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 = ...
10
votes
0answers
284 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
339 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
257 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
213 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 ...
10
votes
0answers
469 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
797 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
915 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
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
249 views

Interesting Integral with Parameters

I would like to compute the following integral: ...
9
votes
0answers
75 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
174 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 ...
9
votes
0answers
205 views

An incorrect answer for an integral

As the authors pointed out in this paper (p. 2), the following evaluation which was in Gradshteyn and Ryzhik (sixth edition) is incorrect (and has been removed). $$ ...
9
votes
0answers
254 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
131 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
222 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 ...
9
votes
0answers
350 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
62 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 ...
8
votes
0answers
123 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 ...
7
votes
0answers
68 views

Prove: $\lim_{h\to0}\int_{-\infty}^{+\infty}|f(x+h)-f(x)|^p\mathrm dx=0$

Suppose for $\forall [\alpha,\beta]\subset(-\infty,+\infty), f\in\mathcal{R}[\alpha,\beta]$, and $\exists ~p>0$, and $\int_{-\infty}^{+\infty}|f|^p\mathrm dx$ exists.Prove: ...
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
93 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
185 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
87 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
165 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 ...
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 ...
7
votes
0answers
98 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
443 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
242 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 = ...
7
votes
0answers
226 views

Connection between integral expression and the factorial of infinity

Does the fact that $$\int_{-\infty}^{\infty}\exp\left(-\frac{1}{2}x^2\right)\mathrm{d}x=\sqrt{2\pi}$$ Have something to do with the fact that the regularized factorial of infinity is also ...
7
votes
0answers
404 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 ...
7
votes
0answers
379 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
77 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] $y(0) = 0$; $t_{0}=0$; $\alpha$, ...