Write $\int_0 ^\frac{\pi}{2}\tan(x)^\alpha\mathrm dx$ as Euler integral.

Question

Write given integrals in form of Euler integrals(Beta,Gamma)

$\int_0 ^\frac{\pi}{2}\tan(x)^\alpha \mathrm dx$ after $\tan x=t$ subst. I get

$\int_0^\infty t^\alpha(1+t^2)\mathrm dt$ which I don't know how convert to Betta or Gamma.

Answer in the book is $\frac{1}{2} \Gamma(\frac{1+\alpha}{2})\Gamma(\frac{1-\alpha}{2})$

Answer 1

You can use the fact (easly obtained via change of variable from the definition) that $$\Gamma(x)\Gamma(y) = 2\Gamma(x+y) \int_{0}^{\pi/2} \cos^{2x-1}\theta \sin^{2y-1}\theta d \theta$$.

In your case $$\Gamma\left(\frac{1-\alpha}2\right) \Gamma\left(\frac{1+\alpha}2\right) = 2\Gamma(1) \int_0^{\pi/2} \cos^{-\alpha}\theta \sin^{\alpha}\theta d\theta,$$ from which you get the result.