# Tagged Questions

Questions involving improper integrals, defined as the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞, or as both endpoints approach limits.

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### 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 ...
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### 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 $\mathbb{H}^4$: ...
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### Evaluating $\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$

Evaluate $$\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$$ I tried using by parts and complex numbers along with series expansion but I was unable to find the answer. Please Help!
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### Juantheron-like integral

While seeing this post, the following integral is just struck me $$\int_0^\infty \frac{dx}{(1+x^2)(1+\tan x)}\tag1$$ I have tried like what user @OlivierOloa did in ...
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### On the change $u=x^{1+\frac{1}{p_n}}$ in $\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx$, where $p_n$ is the nth prime number

In [1] Wikipedia say that for $\Re s>1$ the Riemann zeta function satisfies $$\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx,$$ where $\pi(x)$ is the prime counting function, and say too ...
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### Fourier sine transform of $\frac{1}{2}+\frac{1-x^2}{4x}\ln\vert\frac{1+x}{1-x}\vert$

Show that $$\int_0^{\infty} kF(k)\sin(ka)\,dk = \frac{\pi}{2}aG(a)$$ where $$F(x) = \frac{1}{2}+\frac{1-x^2}{4x}\ln\vert\frac{1+x}{1-x}\vert$$ and $$G(x) = \frac{\sin x-x\cos x}{x^4}$$ EDIT: ...
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### How to find the value of this integral?

This integral to the value \begin{align} \int_0^1\frac{\ln^2(1+x)\ln^2 x}{1-x}\ dx=&\ \color{blue}{-\frac{13\pi^2}{24}\zeta(3)+\frac{47}{2}\zeta(5)-\frac15\ln^52+\frac{\pi^2}9\ln^32-\frac{49\...
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### How to compute or simplify this integration?

Any hints on solving an integration of the following form, $$\int_{x}^{+\infty}\left(1-\frac{1}{1+sy^{-1}}\right) \left(\text{exp}(-\sqrt{y})+ y^{-\frac{1}{2}}(1-\text{exp}(-\sqrt[4]y)\right)dy$$ ...
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### Complicated exponential integral

I encountered the following integral, which I am unable to solve: $$\int_0^\infty {x\over a} e^{{x \over a}} e^{{-b \over a}(e^{{x \over b}}-1)} e^{{-c \over d}(e^{{x \over c}}-1)} \mathrm{d}x$$ ...
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### Closed-form for $\int_0^\infty {\frac{{\ln \left( {1 + x} \right)}}{{1 + ax}}{e^{ - bx}}{x^n}{\rm{d}}x}$

I am trying to find the integration of the following $$\int_0^\infty {\frac{{\ln \left( {1 + x} \right)}}{{1 + ax}}{e^{ - bx}}{x^n}{\rm{d}}x}$$ Here $a>0, b>0$, and $n$ is an integer. I ...
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### Probably Riemann surface integral

Here is the integral: May you please suggest some beautiful idea on using Riemann surface, or some Gauss-Ostrogradsky at the beginning. Also, the initial integral looks really symmetric, so maybe ...
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### Taking an integral of e^(another integral) with respect to the limit of integration of (another integral)

How would I go around integrating $$\int_0^\infty \left( \exp{\left(-\int_{k_0}^{k_0+t} (k_1 + k_2s)(k_3+k_4s)^{s} ds \right) } \right) dt \text{,}$$ where $k_i$ are constants? Is it solvable ...
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### Convergence for a improper integral $\int^b_a fg$

Let $f$ be continuous on [a,b) such that $\int^b_a f$ converges. If $g'$ is locally integrable and has a constant sign on [a,b), prove that $\int^b_a fg$ converges. Edit: We can assume that the limit ...
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### Integral Representation for the Fox-H function on several variables

I have a problem that involves the H-function of several variables, and I have noticed that the implementation of such function when the number of variables are relatively high (greater than 5) must ...
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### A difficult integral $\int_0^{\infty} \frac{\sin 2t}{1+t^3}\, {\rm d}t$

Here is an integral that I want to see a different approach: $$\int_0^{\infty} \frac{\sin 2t}{1+t^3}\, {\rm d}t$$ Well, for someone who is deeply aware of the exponential integral function and the ...
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### How to calculate this Ei(x)-involved definite integral?

I want to solve the integral attached below by means of residue theorem. I tried the common integration ways and seeked references(e.g, Rjadov, et. al). Finally, I decided to solve this integral by ...
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### how to solve this inverse fourier $f(x) =\int^{\infty}_{-\infty} 1/\sqrt{2\pi}\ e^{-2\pi^2/s^2} e^{ i \ s\ x}ds$

I have two functions f(x) and f(s). f(s) is the fourier transform of f(x) and tends to $$e^{-2\pi^2/s^2}$$ I need to take inverse transform of this f(s) to get to f(x). (i need to prove f(x) tends to ...
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### Integrate: $\int\limits_0^\infty{\frac{x^{n-2}}{b\left(1+ ~a x^{\frac{n-1}{n-2}}\right)} \sin{(x b)}~ dx}$

I am trying to solve the integral: $\int\limits_0^\infty{\frac{x^{n-2}}{b\left(1+ ~a x^{\frac{n-1}{n-2}}\right)} \sin{(x b)}~ dx}$ where $x$ is real and $a, b, n$ are positive real constants. I ...
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### on the convergence of a certain integral

If I have an entire function $\phi$ such that it is of exponential order zero. I.e for all $\rho > 0$ we get $|\phi(s)|\le C_\rho e^{|s|^{\rho}}$. Furthermore, I have an extreme decay in the Taylor ...
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### integral involving upper incomplete gamma function

I trying to evaluate the following integral $$\int_0^\infty \dfrac { x^{m-1} \Gamma(A,\mathcal B x^q)} {\left[1+(\eta x)^n\right]^p} \,\mathrm dx$$ where the integration is w.r.t. $x$, and the ...
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### Determine the behavior of a function defined by an integral

Suppose we have a function defined by $$\varphi(s)=\int_{-\infty}^\infty f(x,s)\,dx$$ defined for $s\in S\subseteq \mathbb{R}$. Suppose we know that it blows up at $a\in \partial S$, and we want to ...
Let $J_\nu(x)$ be the Bessel function of the first kind. $\int_0^\infty J_\nu(x)dx=1 , (Re(\nu)>-1)$. $\lim_{\nu\to+\infty}J_\nu(x)=0$ for any fixed $x$. I think the above two properties of ...
### Integral $\int_{0}^\infty\frac {(1-{{e}^{-i (q-p)t}})ln(|p^2-p_0^2|)}{(q-p)({{ p}}^{2}-{{p_1}}^{2})({{p}}^{2}-{{p_2} }^{2})}dp$
I am trying to get a closed form analytic result for the integral \int _{0}^{\infty }\!{\frac {\left(1-{{\rm e}^{-i \left( {q}-{p} \right) t}}\right){\rm ln}(|p^2-p_0^2|)}{ ( {q}-{p} ) \left( {{ p}...