# Tagged Questions

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### An integral similar to integrating the Beta function

Peace be upon you, I have the following definite integral for the mathematical expectation of some distribution \begin{align*} \int_{-1}^1 z^2\int_{\max(z-1,-1)}^{\min(0,z)} (z-y)^a(-y)^b \ ,dy \,dz ...
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### Integral of $\ln(x)\operatorname{sech}(x)$

How can I prove that: ...
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### Solving integral that contain exponential function and lower incomplete gamma function

I have the following integral; $$y=\int_0^\infty\frac{e^{-xf}}{m+x}\gamma(a,hx)~dx$$ where $f,m,h\in\mathbb{R}^+$ , $a\in\mathbb{N}$ , $\gamma\left(a,h x\right)$ is the lower incomplete gamma ...
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### Scary contour integral, but is also an integral representation for $\Gamma$-function

This problem is supposed to be from an old Acta Mathematica volume I circa 1880's, and is attributed to Bourguet. By using a parabola with its focus on the origin as a contour, show that: ...
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### Evaluation of the definite integral: $\int_0^{\pi/2}x\tan(x)^pdx$

The integral: $$J=\int_{0}^{\pi/2}x\,\tan^{p}\left(x\right)\,{\rm d}x$$ has the solution: ...
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### Integral involving Gamma Function

I am solving the following integral: $$\int_{-1}^{K} u^B e^{-u} du$$ The solution of the integral is a lower incomplete Gamma Function if -1 is replaced with 0. Can anybody help me in solving the ...
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### Evaluating $\dfrac{1}{\Gamma (r)}\int_{0}^{x}(x-t)^{\alpha -1}t^{\lambda}dt$ [closed]

How can I evaluate the following integral $$\frac1{\Gamma(r)}\int_0^x(x-t)^{\alpha-1}t^\lambda\ dt$$
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### Compute $\int_{0}^{\infty}e^{-tz}(z+d)^{n-1}dz$ as a function of $\Gamma(n)$

Is it possible to compute this integral $$\int_{0}^{\infty}e^{-tz}(z+d)^{n-1}dz$$ as a function of complete gamma $\Gamma(n)$. If possible, I'm looking for a closed form solution. Thanks!
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### Evaluating improper integral expression

Can anybody please guide me in evaluating this expression, for my research work. I have tried a lot but in vain. Both the integrals involve gamma functions and even wolfram said that the time limit ...
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### Prove $\int_{0}^\infty \frac{1}{\Gamma(x)}\, \mathrm{d}x = e + \int_0^\infty \frac{e^{-x}}{\pi^2 + \ln^2 x}\, \mathrm{d}x$

I came across this nice identity: $$\int_{0}^\infty \frac{1}{\Gamma(x)}\, \mathrm{d}x = e + \int_0^\infty \frac{e^{-x}}{\pi^2 + \ln^2 x}\, \mathrm{d}x$$ Is there an elementary proof?
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### integral involving incomplete gamma function

Need to evaluate the integral $$\int_a^b e^{1/x}\,\Gamma(m,1/x)\,dx$$ or equivalently $$\int_{1/a}^{1/b} y^{-2}\,e^{y}\,\Gamma(m,y)\,dy,$$ where $m$ is an integer, and $0<a<b<\infty$. The ...
I am trying to integrate this: $$\int_0^\infty z^{-|M|-1}\,\Gamma(A,z)\;dz$$ where $A$ is a real positive, and note that the power of $z$ is $-|M|-1$, i.e., is forced to be negative real.