# Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

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### Can someone explain this integration trick for log-sine integrals?

I was working on this rather challenging log-sine integral: $$\int_{0}^{2\pi}x^{2}\ln^{2}\left(2\sin\left(x \over 2\right)\right)\,{\rm d}x = {13\pi^{5} \over 45}$$ The upper limit is a waiver ...
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### Evaluate $\int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}dx$

I need the method to evaluate this integral (the closed-form if possible). $$\int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}\,dx$$ I used the relationship between $\tan x$ and $\tanh x$ but it didn't ...
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### Evaluating $\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}\,\mathrm{d}x$

Evaluate $$\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}\,\mathrm{d}x.$$ 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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### On the relationship between $\Re\operatorname{Li}_n(1+i)$ and $\operatorname{Li}_n(1/2)$ when $n\ge5$

Motivation $\newcommand{Li}{\operatorname{Li}}$ It is already known that: $$\Re\Li_2(1+i)=\frac{\pi^2}{16}$$ $$\Re\Li_3(1+i)=\frac{\pi^2\ln2}{32}+\frac{35}{64}\zeta(3)$$ And by this question, ...
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### Omega Constant Integral

Whilst reading this Math SE post, I saw that the OP mentioned the integral $$\int_0^\infty \frac{1+2\cos x+x\sin x}{1+2x\sin x +x^2}dx=\frac{\pi}{1+\Omega}$$ where $\Omega$ is the unique solution to ...
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### Double integral - transformation

I'm trying to calculate $$\iint_{\Omega } e^{(x+y^2)^{3/2}} \,\mathrm{d}A,$$ where $$\Omega =\{x,y>0 : x+y\leq 2\}.$$ Not sure where to go with it. I need to find a transformation and then ...
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### An extremely mysterious integral: $\int_0^1 \frac{k \tan^{-1}(t)}{k^2 + t^2}\mathrm d t$

$$f(n) = \int_0^1 \frac{n \tan^{-1}(t)}{n^2 + t^2}\mathrm d t \tag{n > 2}$$ Introduction: This is one of the most beautiful and mysterious integrals I've every encountered. It's very simple, but ...
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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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### Help with an unusual integral

Let $n$ be a positive integer, and $s\in \mathbb{C}\;,\Re(s)>0$. I want to compute the integral : $$\int_{0}^{\infty}\sin\left(2\pi ne^{x}\right)\left[\frac{s}{e^{sx}-1}-\frac{1}{x}\right]dx$$ I ...
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### Average value of $\ln(1+e^x)$ when $x$ is normally distributed

Does the following integral admit a closed form answer: $$\int_{-\infty}^\infty\mathrm d x \exp\left(-\frac{(x-\mu)^2}{2\nu}\right) \ln(1+e^x)$$ where $\nu>0$ and $\mu$ are finite real parameters....
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### Integrating $\int_{0}^{\infty} \frac{p^6 dp }{1 + a p^4 + b p^6 } \int_{0}^{\pi}\frac{\sin^5 \theta \,d\theta}{1 + a |p-k|^4 + b |p-k|^6 }$

This is my first question here, so I hope I'm not giving too little/too much information. I need some help calculating (or even approximating) an integral which I've been wrestling with for a while. ...
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### 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?) ...
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### Exact values of error function

The error function is defined as $$\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} \, dt.$$ We know that the Gaussian integral is $$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$ ...
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### Closed form of $\int_0^1\frac{W_0(-t/e)}{W_{-1}(-t/e)} \,dt$

$\require{begingroup} \begingroup$ $\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}\def\Ei{\operatorname{Ei}}$ Is there a known closed form for ...
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### Closed form for the integral $\int_{-\pi/2}^{\pi/2} W(\sec(\varphi)) d\varphi$?

I'm interested in the integral $$I=\int_{-\pi/2}^{\pi/2} W(\sec(\varphi)) d\varphi$$where $W$ is the Lambert W function, defined such that $W(x)e^{W(x)}=x$. The numerical value for $I$, as yielded by ...
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### The sine cardinal function and $F_1 = F_2 = F_3 = F_4 = F_5 = F_6 = 0$

Define the function, $$F_n=\frac12-\int_0^\infty \frac{\sin^n x}{x^n}\,dx+\sum_{x=1}^\infty \frac{\sin^n x}{x^n}\tag1$$ where $\rm{sinc}^n(x)=\frac{\sin^n x}{x^n}$ is the sine cardinal function. We ...
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### Methods to solve $\int_{0}^{\infty} \frac{\cos\left(kx^n\right)}{x^n + a}\:dx$

Spurred on by this question, I decided to investigate for different functions on the numerator. Here, I went from $\exp(..)$ to $\sin(..) / \cos(..)$. I initially thought I could modify the result ...
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### How prove this integral $\int_0^1\frac{(\arctan{x})^2\ln({x+1/x+2})}{(1+x)^2}dx$

$$I=\int_0^1\frac{(\arctan{x})^2\ln({x+1/x+2})}{(1+x)^2}dx=-\dfrac{\pi^3}{96}+\dfrac{5\pi}{16}\ln^22-\dfrac{\pi}{4}G-G+\dfrac{\pi}{2}\ln2+\dfrac{7}{16}\zeta(3)$$ Where G is the Catalan's Constant. ...
### Is there a closed form for $\,_4 F_3(1,1,1,3; 3/2,5/2,5/2;1)$?
A semi-algebraic generalization of the Steiner surface has appeared, $$S = \left\{(x,y,z,t) \space \vert \space t^2(1-x^2-y^2-z^2-t^2) - (x^2 y^2 + x^2 z^2 + y^2 z^2 - 2 x y z) \geq 0 \right\}$$ ...