All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

learn more… | top users | synonyms (2)

27
votes
0answers
300 views
+100

Ramanujan log-trigonometric integrals

I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$ R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} ...
14
votes
0answers
531 views

The log gamma integral $\int_{0}^{z} \log \Gamma (x) \ \mathrm dx$

One way to evaluate $ \displaystyle\int_{0}^{z} \log \Gamma(x) \ \mathrm dx $ is in terms of the Barnes G-function. $$ \int_{0}^{z} \log \Gamma(x) \ \mathrm dx = \frac{z}{2} \log (2 \pi) + ...
13
votes
0answers
54 views

Computing $\lim\limits_{n\to\infty} \Big(\sum\limits_{i = 1}^n \sum\limits_{j = 1}^n \frac1{i^2+j^2}-\frac{\pi}{2} \log(n)\Big)$.

In the chatroom we discussed about the asymptotic of $\displaystyle \sum_{i = 1}^n \sum_{j = 1}^n \frac1{i^2+j^2}$, and if we think of the inverse tangent integral, it's easy to see that ...
13
votes
0answers
444 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 ...
13
votes
0answers
503 views

Closed form for this integral $\int_{0}^{\infty}\frac{dx}{\sqrt{x}}\, e^{-x^{2}-\frac{b^{2}}{x}}$

How would you evaluate this integral? \begin{equation}\int_{0}^{\infty}\frac{dx}{\sqrt{x}}\, e^{-x^{2}-\frac{b^{2}}{x}}\end{equation} It reminds me of the form of a modified Bessel function of the ...
11
votes
0answers
214 views

The closed form of $\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$

What tools, ways would you propose for getting the closed form of this integral? $$\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$$
10
votes
0answers
252 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: $$x_1^2 + \dots + x_i^2 + \dots + x_n^2 = 1,$$ bounded by an ellipsoid $$a_1x_1^2 + \dots + a_ix_i^2 + \dots + ...
9
votes
0answers
122 views

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

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 $$ ...
9
votes
0answers
292 views

Integral $\int_0^\infty \frac{\log^2 x \cos ax}{x^n-1}dx$

Hi I am trying to calculate $$ I:=\int\limits_0^\infty \frac{\log^2 x \cos (ax)}{x^n-1}\mathrm dx,\quad \Re(n)>1, \, a\in \mathbb{R}. $$ Note if we set $a=0$ we get a similar integral given by $$ ...
9
votes
0answers
137 views

A closed form for $\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx,\ a\notin\mathbb{Z}^+$

Let $$I(a)=\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx.$$ $I(a)$ has closed form representations for all $a\in\mathbb{Z}^+$. Is there any algebraic (or at least period) ...
9
votes
0answers
560 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
353 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: ...
8
votes
0answers
56 views

Is there a name for the class of functions which are infinitely integrable in elementary functions?

Is there a name for the class of functions which are infinitely integrable in elementary functions, that is whose consecutive integrals also elementary not depending on how much times we took the ...
8
votes
0answers
175 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 ...
8
votes
0answers
151 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 ...
8
votes
0answers
134 views

Can a substitution cause a convergent definite integral to diverge?

Is it possible for an integral, $$\int_a^b f(x)\text{d}x,$$ to converge, but for some substitution $x=g(y)$ to cause the integral, $$ \int_{g^{-1}(a)}^{g^{-1}(b)} f(g(y)) g'(y) \text{d}{y},$$ to ...
8
votes
0answers
157 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 ...
8
votes
0answers
398 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 ...
8
votes
0answers
574 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 ...
7
votes
0answers
99 views

Evaluating $\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \ dx$

What tools would you recommend me for evaluating this integral? $$\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \ dx$$ My first thought was to use beta function, but it's hard to get ...
7
votes
0answers
322 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
171 views

How to evaluate the integral $e^{-(c\ln(\frac{1}{x}))^s} dx$?

Can anyone help me evaluate $$\int_{\alpha}^1 \exp{\left\{-\left(c\ln\left(\frac{1}{x}\right)\right)^s\right\}} dx$$, Where $0 \leq \alpha \leq 1$ and $s \in \mathbb{R}$. I tried changing ...
7
votes
0answers
271 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
184 views

Definite integral $\int_0^{2\pi}\frac{ab}{\sqrt{b^2\cos^2(\theta)+a^2\sin^2(\theta))}}\cos^2(\theta) d\theta$

Could you help me finding the following definite integral, with $a$ and $b$ constants? Thank you! $$\int_0^{2\pi}\frac{ab}{\sqrt{b^2\cos^2(\theta)+a^2\sin^2(\theta))}}\cos^2(\theta) d\theta$$
7
votes
0answers
135 views

Closed-form expression for an iterated integral

Does the following iterated integral have a simple closed-form expression in terms of $z$? $$ I = \int_0^\infty \int_0^\infty \sqrt{\frac{1 + x^2 y^2 + x^2 z^2 + y^2 z^2}{(x^2 + y^2 + z^2 + x^2 y^2 ...
7
votes
0answers
182 views

Evaluating $\int_0^\infty \frac{1}{x+1-u}\cdot \frac{\mathrm{d}x}{\log^2 x+\pi^2}$ using real methods.

By reading a german wikipedia (see here) about integrals, i stumpled upon this entry 27 1.5 $$ \color{black}{ \int_0^\infty \frac{1}{x+1-u}\cdot \frac{\mathrm{d}x}{\log^2 x+\pi^2} ...
6
votes
0answers
152 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 ...
6
votes
0answers
121 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). $$ ...
6
votes
0answers
172 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 ...
6
votes
0answers
84 views

Asymptotic property of integral involving Bessel function.

Consider the following integral $$ I(s)=\int_{0}^{\infty}{J_{\frac{n-2}{2}}}(sr)r^{A+1}(e^{-r^{2\alpha}}-1)dr, $$ where $J_{\frac{n-2}{2}}$ is the Bessel function of order ${\frac{n-2}{2}}$, $s, A, ...
6
votes
0answers
277 views

Surface integral for a scalar function defined on a discrete surface

Imagine a polyhedral, discrete surface embedded in $\mathbb{R}^3$. Its faces are all triangles. For each vertex, one can compute the discrete mean and Gaussian curvatures and evaluate the sum of ...
6
votes
0answers
281 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 ...
6
votes
0answers
214 views

Integral of a gaussian function of trigonometric functions

I need help with the analytical solution of this integral: ...
6
votes
0answers
65 views

Various integration theories

Could anyone briefly explain, or point me towards a resource explaining, the main differences between the main integration theories, namely: Riemann Integration Riemann-Stieltjes Integration ...
6
votes
0answers
275 views

To determine if$f^{-1}(x)$ is periodic function or not? $f(x)=\int_1^{x} \frac{1}{\sqrt[m]{P(t)}}\;dt$

$$f(x)=\int_1^{x} \frac{1}{\sqrt[m]{P(t)}}\;dt$$ $P(x)$ is polynomial with degree $n$. $m$ is an positive integer and $m>1$ What is the algoritm to determine $f^{-1}(x)$ is periodic function ...
6
votes
0answers
144 views

Evaluting $ \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 $$ \int_0^{\infty}\dfrac{v}{\sqrt{v + c}}e^{-\dfrac{y^2}{2(v + c)} - \dfrac{(u-v)^2}{u^2v}}dv,$$ ...
6
votes
0answers
591 views

Nasty Integral - Closed form solution?

Any suggestions on how to integrate this beast?: $$\int_0^{\omega_t}\int_{\omega_t}^f\sin^2(\omega_{12}/2)\sin^2(\omega_{23}/2)d\omega_{23}d\omega_{12}$$ where: $f = 2\pi+2\tan^{-1}(y,x)$ $y = ...
5
votes
0answers
91 views
+200

Closed form for integral $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $

I'm looking for a closed form of this definite iterated integral. $$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $$ From Vladimir ...
5
votes
0answers
69 views

Wicked domain of integration in a triple integral

I am dealing with a domain of integration of the form: $\left(\frac{x-y}{x+y}\right)^2+\left(\frac{y-z}{y+z}\right)^2+\left(\frac{x-z}{x+z}\right)^2\leq k$ The region looks like this (for $k=0.2$): ...
5
votes
0answers
47 views

How to compute product integrals?

From the wikipedia article about product integrals I can see that if our function is scalar, then to compute type I product integral we can just take exponential of a usual integral: $$\prod_a^b ...
5
votes
0answers
51 views

Integration of bundle-valued differential forms

The literature, at least textbooks, seems to be very scarce on the topic of integrating bundle-valued differential forms. So I wonder where can I read on the topic? I want to see usual theorems, like ...
5
votes
0answers
73 views

Functional inequality with a strong RHS

Consider a continuous function $f:[0,1]\to\mathbb{R}^{+}$. Show that $$\int_0^1 f(x)dx-\exp\left(\int_0^1 \log(f(x)) dx\right)\le \max_{0\le x,y\le 1}\left(\sqrt{f(x)}-\sqrt{f(y)}\right)^2$$ I ...
5
votes
0answers
64 views

Is $\sigma$-finiteness really a necessary condition for this problem?

Question: Let $(X, \mathcal A, \mu)$ be a measure space and suppose $\mu$ is $\sigma$-finite. Suppose $f$ is integrable. Prove that given any $\varepsilon$, there exists a $\delta >0$ such that ...
5
votes
0answers
64 views

An inequality between integrals of series of characteristic functions of cubes

Let $1\leq p<\infty$. Prove that there exists $C>0$ such that $$ \left(\int\left|\sum_{i=1}^\infty a_i\chi_{2Q_i}\right|^p \, dx\right)^{1/p} \leq C\left(\int\left|\sum_{i=1}^\infty ...
5
votes
0answers
34 views

A p-adic integral

Let $(K,||)$ be a finite extension of $\mathbb{Q}_p$ of degree $d$ such that the restriction of $||$ to $\mathbb{Q}_p$ is the usual p-adic absolute value. Endow $GL_n(K)$ with the unique Haar measure ...
5
votes
0answers
72 views

Relations between definite integrals not having a known closed form

Are there any known cases, when there are two (or more) definite integrals, none of them having any known closed-form expression on its own, but there is still a non-trivial$^\dagger$ elementary ...
5
votes
0answers
133 views

Showing some complicated integral expression is bounded

In my research, I come across some expression I need to bound. I wish to show that the following integral is bounded in $t$ and $x$, i.e. the following supremum is finite: $$\sup_{t,x\in ...
5
votes
0answers
147 views

Why is Lebesgue-Stieltjes a generalization of Riemann-Stieltjes? Moreover, is there an example where Lebesgue-Stieltjes is useful

I certainly have a question, but i don't know what the best title should be. Please edit the title if there is a better one :) And I believe, to get a better answer, it would be good to explain ...
5
votes
0answers
52 views

Leibniz integral rule implementation

Can someone please explain to me why the following expression is true? I really tried to figure out how Leibniz integral rule works, but everytime I think I managed to figure out how to implement it, ...
5
votes
0answers
1k views

How prove this hypersingular operator $(T\psi)((z(t))$

Let $\Omega\subset R^2$ be a simply connected bounded domain with infinitely differentiable boundary $\partial\Omega$and unit normal vector $v$ directed into the exterior of $\Omega$ ...