# Tagged Questions

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### Evaluation of the integral $\int_0^1 \log{\Gamma(x+1)}\mathrm dx$

As it says in the title, I'd like to know how to solve the definite integral $\int_0^1 \log{\Gamma(x+1)}\mathrm dx$. Mathematica gives the answer $\frac{1}{2}\log (2\pi)-1$ but I have no idea how one ...
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### How to solve gamma function integral for 4!

I have been trying to test my knowledge of the gamma function by calculating 1! and 4!. I got the right result for 1! but I cannot get 4! analytically. To be more specific, I think I am not ...
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### Some more parameter integrals

It seems that the following formulas hold : $$\int_{0}^{\infty} \frac{1}{\sqrt{x^{2n}+1}} dx = \frac{\Gamma(\frac{n-1}{2n})\Gamma(\frac{2n+1}{2n})}{\sqrt\pi}$$ for any integer $n > 1$ and ...
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### $\Gamma ( \alpha)$ function

Gamma function of $\alpha$ is defined as $\Gamma \left( \alpha \right) = \int\limits_0^\infty {y^{\alpha - 1} e^{ - y} dy}$ Gamma function exist for $\alpha > 0$ why??? I think the reason is ...
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### Strategy for Improper Integrals Related to the Beta Function 2

I am looking for the solution of the following integral $$\int_0^1 y^k \log\left(1+\left(\frac y{1-y}\right)^a\right)dy,\quad a>0$$ I really appreciate it if any one can help.
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### Help evaluating a gamma function

I need to do a calculus review; I never felt fully confident with it and it keeps cropping up as I delve into statistics. Currently, I'm working through a some proof theory and basic analysis as a ...
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### How to calculate these gamma functions?

Equation : $$\int _{0}^{\infty }x^{n}e^{-x}dx=n!=\Gamma(n+1)$$ 1) $$\int _{0}^{1}x^{2}\left( \ln \dfrac {1} {x}\right) ^{3}dx$$ 2) $$\int _{0}^{1}\sqrt[3] {\ln x}dx$$ Hint : $$x=e^{u}$$ ...
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### Integrating $\int_0^{\infty} u^n e^{-u} du$

I have to work out the integral of $$I(n):=\int_0^{\infty} u^n e^{-u} du$$ Somehow, the answer goes to $$I(n) = nI(n - 1)$$ and then using the Gamma function, this gives $I(n) = n!$ What I do ...
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What is the integral or maybe a suggested technique to find $$\int_0^x \left(1+\left(\frac{t}{b}\right)^a\right)^{-q} d t,$$ $a,b,q\in \mathbb{R}:a\cdot q>1,b>0.$ ?
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### $\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$

I'm trying to solve the integral $\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$, where $s$, $r$ and $m$>1 are positive integers. My question is whether a closed form ...
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### Solve in terms of the Gamma function

Show: \begin{align*} \int\limits_0^1\sqrt{\frac{1-x^2}{1+x^2}}\,\mathrm d x &=\frac{\sqrt \pi}{4}\left(\frac{\Gamma ...
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### what are the properties of gamma function? [closed]

In mathematics, the gamma function (represented by the capital Greek letter $\Gamma$) is an extension of the factorial function, example: $\Gamma(x)$, $\Gamma(ix)$. What are the physical properties ...
What are the usual ways to follow in order to solve the integrals given below? \begin{align*} I&=\int_0^1 \ln\Gamma(x)\,dx\\ J&=\int_0^1 x\ln\Gamma(x)\,dx \end{align*}
I'm trying to prove that for $p,q>0$, we have $$\int_0^1t^{p-1}(1-t)^{q-1}\,dt=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}.$$ The hint given suggest that we express $\Gamma(p)\Gamma(q)$ as a double ...