# Tagged Questions

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### Integrals of integer powers of dilogarithm function

I'm interested in evaluating integrals of positive integer powers of the dilogarithm function. I'd like to see the general case tackled if possible, or barring that then as many particular cases as ...
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### Integral $\int_{0}^1 \sqrt{\frac{\ln{x}}{x^2-1}} dx$

Please help evaluating this integral $$\large\int_{0}^1 \sqrt{\frac{\ln{x}}{x^2-1}} dx$$ Mathematica could not evaluate it in a closed form. Numerically it is about ...
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### Integral/infinite sum related to Bessels which pop up in optical coherence theory

In propagating partially coherent optical fields, the following integral pops up: $$I_1=\int_0^{2\pi} e^{i(a\cos[\theta]+b\cos^2[\theta])}d\theta,$$ where $a$ and $b$ are real numbers. If we ...
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### Examples of pairs of difficult integrals

I’m looking for pairs of difficult definite integrals that are linked algebraically on a certain field without known change of variable or integration by parts from one integral to the other. Here a ...
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### Can we express the following in a closed form? [duplicate]

I want to evaluate the integral: $$I=\int_{0}^{\pi/2}\ln \left ( \frac{(1+\sin x)^{1+\cos x}}{1+\cos x} \right )\,dx$$ Well, the sub $u=\pi/2-x$ does not give me any result. In fact it makes the ...
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### closed form for $\int_0^1x^{a+1}(1-x^2)^bJ_a(cx)dx$

$$\int_0^1x^{a+1}(1-x^2)^bJ_a(cx)dx$$ my friend posted the integral on the fb and I tried to solve it but I faild because I have little information about bessel function so could some one help ?
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### Evaluate $\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx$

Is there a closed form for the integral $$\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx?$$ where $\lambda>0$, $a>0$, $d>0$ and where $b$, ...
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### Closed form for $f(x)=\int_{0}^{+\infty}e^{it^{x}}dt$?

Let $x>1$ and $f(x)=\int_{0}^{+\infty}e^{it^{x}}dt$. Does this integral have a closed form ? Fist point, the integral converges. Indeed let $u=e^{it^{x}}$ and $v=\frac{-i}{x}t^{1-x}$ we have ...
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### Definite integral - closed form: $\int_{0}^{\infty}\cos\left(x^{4} + 1 \over x^{2}\right)\,{\rm d}x$

I'm struggling with this definite integral: $$\int_{0}^{\infty}\cos\left(x^{4} + 1 \over x^{2}\right)\,{\rm d}x.$$ Any help will be greatly appreciated.
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### Does this integral have any closed form? $\displaystyle\int\frac{1}{x+\sin(x+1)}\mathop{\mathrm dx}$

Does this integral have any closed form? $$\int\frac{1}{x+\sin(x+1)}\mathop{\mathrm dx}$$ I think the substitution $x=(u-1)+2\pi$ will do it, no?
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### Integration of $\int_0^\pi\int_0^\pi[\sin x\sin y\ge a]\,dx\,dy$

Does the integral $$I(a)=\int_0^\pi\int_0^\pi[\sin x\sin y\ge a]\,dx\,dy$$ have any closed-form solution? This is the area under the contours of $\sin x\sin y$; $[\cdot]$ is the Iverson bracket, which ...
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### Closed form for $\int_0^{\pi/2}\frac{\sqrt{1+\sin\phi}}{\sqrt{\sin2\phi}\,\sqrt{\sin\phi+\cos\phi}}d\phi$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^{\pi/2}\frac{\sqrt{1+\sin\phi}}{\sqrt{\sin2\phi} \,\sqrt{\sin\phi+\cos\phi}}d\phi$$ Its approximate numeric value is ...
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### How to find this integral $\int_{0}^{1}\frac{x}{1-x^4}\arctan{\frac{x-x^5}{1+x^6}}\,dx$

Find the integral value $$I=\int_{0}^{1} {x \over 1 - x^{4}}\,\arctan\left(x - x^{5} \over 1 + x^{6}\right)\,{\rm d}x$$ My good friends gave me this problem, and I can't solve it. Using computer I ...
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### Need help with $\int_0^\infty\frac{e^{-x}}{\sqrt[3]2+\cos x}dx$

Please help me to evaluate this integral: $$\int_0^\infty\frac{e^{-x}}{\sqrt[3]2+\cos x}dx$$
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### Closed form for $\int_{-\infty}^0\operatorname{Ei}^3x\,dx$

Let $\operatorname{Ei}x$ denote the exponential integral: $$\operatorname{Ei}x=-\int_{-x}^\infty\frac{e^{-t}}tdt.\tag1$$ It's not difficult to find that ...
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### Integral $\int_{0}^{3} \frac{e^{-x^2}}{\sqrt{1-x^2}} \,dx$

I recently got stuck on evaluating the following integral, $$\int_{0}^{3} \frac{e^{-x^2}}{\sqrt{1-x^2}} \,dx.$$ Is it possible to evaluate this integral in a closed form? I am not sure if there is ...
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### Evaluating $\int_0^{\frac{1}{2}}\log^2(2\sin(\pi x))\cos(\pi(1-2x))dx$ [closed]

How can we prove following formulas $$\int_0^{\frac{1}{2}}\log^2(2\sin(\pi x))\cos(\pi(1-2x))dx=\frac{-1}{4}$$ or $$\int_0^{\frac{1}{2}}\log^3(2\sin(\pi x))\cos(\pi(1-2x))dx=\frac{\pi^2+6}{16}$$
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### Closed form for $\int_{0}^{1/2}\left(2x - 1\right)^{6}\ \log^{2}\left(2\sin\left(\pi x\right)\right)\,{\rm d}x$

How can I find a closed form for the following integral $$\int_0^{1/2}\left(2x - 1\right)^{6}\ \log^{2}\left(2\sin\left(\pi x\right)\right) \,{\rm d}x$$
### Closed form of $\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx$
What is the closed form of the following integral $$\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx,n\in\mathbb{N}$$ By Mathematica I saw that $$\int_0^\frac{1}{2}x\cot(\pi x)\,dx=\frac{\log(2)}{2\pi}$$ ...
### Integral $\int_0^\infty\exp\left(-\sqrt2\,x^2\right)\,\operatorname{erfi}(x)\,\log(x)\,x^3\,dx$
Consider the following integral: $$\mathcal{A}=\int_0^\infty\exp\left(-\sqrt2\,x^2\right)\,\operatorname{erfi}(x)\,\log(x)\,x^3\,dx,\tag1$$ where $\operatorname{erfi}(x)$ denotes the imaginary error ...