Expressing $\int_{-\infty}^\infty dx/(x^2+1)^n$ in terms of Gamma function

Question

How to prove this identity for $n>1/2$? $$\int_{-\infty}^{\infty}\frac{dx}{(x^2+1)^n}=\frac{\sqrt{\pi}\cdot \Gamma(n-\frac{1}{2}) }{\Gamma (n)}$$

Answer 1

Hint: Due to parity, $\displaystyle\int_{-\infty}^\infty f(x)~dx=2\cdot\int_0^\infty f(x)~dx.~$ Then, let $t=\dfrac1{1+x^2}$ , and recognize the expression of the beta function in the new integral.