Express in gamma/beta functions $\int_{0}^{\pi/2}\tan^n x \ dx$

Question

Express in gamma/beta functions and find n's that intergral converges.

$\int_{0}^{\pi/2}\tan^n x \ dx$

i am stuck a little bit

Answer 1

Hint:

Let $t=\sin^2x$, giving $dt=2\sin x\cos x\,dx$.

Then

$$2\int \sin^n(x)\cos^{-n}(x)\,dx=\int \sin^{n-1}(x)\cos^{-n-1}(x)\,dt=\int t^{(n-1)/2}(1-t)^{-(n+1)/2}\,dt.$$