# Tagged Questions

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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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### Integral $\int_0^1 \log \left(\Gamma\left(x+\alpha\right)\right)\,{\rm d}x=\frac{\log\left( 2 \pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha$

Hi I am trying to prove$$I:=\int_0^1 \log\left(\,\Gamma\left(x+\alpha\right)\,\right)\,{\rm d}x =\frac{\log\left(2\pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha\,,\qquad \alpha \geq 0.$$ I am ...
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### Integral calculus question relating to particle motion

"A particle of mass m is attracted toward a fixed point 0 with a force inversely proportional to its instantaneous distance from 0. If the particle is released from rest, at distance L, from 0, find ...
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### Integral of incomplete gamma function

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.
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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 ...
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 ...
Let $a_n = \int\limits_{0}^{n} e^{-x^4} dx$. Does $\{ a_n \}_{n \rightarrow \infty}$ converge? $\{ a_n \} =\{ \int\limits_{0}^{1} e^{-x^4} dx, \int\limits_{0}^{2} e^{-x^4} dx, ..., ... 6answers 204 views ### How to integrate$\int_{0}^{a}x^{n-1}e^{-x}dx$We know that $$\int_{0}^{\infty }x^{n-1}e^{-x}dx = \Gamma (n)$$ But how do we integrate this? $$\int_{0}^{a}x^{n-1}e^{-x}dx$$ 1answer 101 views ### Interesting integral involving$\Gamma (z)$. Find the value of $$\int_0^\infty t^{x-1}e^{-\lambda t \cos(\theta)} \cos(\lambda t \sin (\theta)) dt$$ where$\lambda >0$,$x>0$, and${-1\over 2}\pi < \theta < {1\over 2}\pi$in terms of ... 1answer 85 views ### Expressing integral as gamma function I was trying to compute the expected value$E[X^k]$for a Weibull distribution, and I encountered this integral $$\int_{-\infty}^\infty t^{b+k-1}e^{-ct^b}dt$$ where$k>0$is an integer and ... 1answer 114 views ### Gamma function in the sight of Lebesgue and Riemann integration. I am taking a somewhat hard measure theory course and I was asked to prove this: a) Let$\alpha > 0$be a real number. Prove that $$\Gamma(\alpha):=\int_0^\infty e^{-x}x^{\alpha-1}dx$$ exists. ... 0answers 469 views ### The log gamma integral$\int_{0}^{z} \log \Gamma (x) \ \mathrm dx$One way to evaluate$ \displaystyle\int_{0}^{z} \log \Gamma(x) \ \mathrm dx $is in terms of the Barnes G function. $$\int_{0}^{z} \ln \Gamma(x) \ \mathrm dx = \frac{z}{2} \log (2 \pi) + ... 2answers 137 views ### Gamma Type Integral I was hoping someone could help me with a question I came across recently: essentially it's a gamma type integral that your asked to evaluate/reduce: ... 0answers 108 views ### Limiting behavior of an integral involving incomplete Gamma function I am wondering about the limiting behavior as k\rightarrow\infty of the following integral:$$I(k)=\frac{2^{-k/2}}{\Gamma(k/2)}\int_{f(k)}^\infty ... 1answer 31 views ### Relation between two gamma functions Does anyone know the relation between these two gamma functions? 1st) Gamma[1 + c, a (1 + b)] 2nd) Gamma[c, a (1 + b)] The question is: may I write the 1st like the 2nd times something? thank you ... 0answers 33 views ### Show that$\sum\limits_{k=0}^{y-1}(-1)^k\frac{\binom{y-1}{k}}{k+x}=\frac{(x-1)!(y-1)!}{(x+y-1)!}$. [duplicate] Prove that for$x,y$positive integers, $$\sum_{k=0}^{y-1}(-1)^k\frac{\binom{y-1}{k}}{k+x}=\frac{(x-1)!(y-1)!}{(x+y-1)!}$$ One way is to use the beta-gamma functions relation: ... 3answers 164 views ### I'm looking for several ways to prove that$\int_{0}^{\infty }\sin(x)x^mdx=\cos(\frac{\pi m}{2})\Gamma (m+1)$I'm looking for several ways to prove that $$\int_{0}^{\infty }\sin(x)x^mdx=\cos(\frac{\pi m}{2})\Gamma (m+1)$$ for$-2< Re(m)< 0$2answers 87 views ### Using$\Gamma(z) = \lim_{n \to \infty} \left[ \frac{n! n^z}{z(z+1)(z+2)\ldots(z+n)} \right]$to prove the Weiestrass product. I was searching the web quite thoroughly in the last two days. I was in paralytically looking for a rigorous proof using $$\Gamma(z) = \lim_{n \to \infty} \left[ \frac{n! ... 5answers 499 views ### Show that \int_{0}^{\infty }\frac {\ln x}{x^4+1}\ dx =-\frac{\pi^2 \sqrt{2}}{16} I could prove it using the residues but I'm interested to have it in a different way (for example using Gamma/Beta or any other functions) to show that$$\int_{0}^{\infty }\frac {\ln x}{x^4+1}\ dx ... 1answer 132 views ### Integrate the beta function$B(x,y)$over the unit square. I was looking a bit around on the site and disovered (to my surprise) that $$\int_0^1 \log \Gamma(x+t)\,\mathrm{d}x = \log \Bigl(\sqrt{2\pi}\Bigr) - t + t \log t$$ and using this and the fact ... 7answers 638 views ### Evaluating$\int_0^\infty \frac{dx}{1+x^4}$. [duplicate] Can anyone give me a hint to evaluate this integral? $$\int_0^\infty \frac{dx}{1+x^4}$$ I know it will involve the gamma function, but how? 1answer 147 views ### Integration gamma and beta:$\int_0^4y^3\sqrt{64-y^3}\,\mathrm dy\$
How can we evaluate the following integral? $$\int_0^4y^3\sqrt{64-y^3}\,\mathrm dy$$ I can't find anything to substitute because all of the trigonometric identities are in square form...