How to prove this identity for $n>1/2$? $$\int_{-\infty}^{\infty}\frac{dx}{(x^2+1)^n}=\frac{\sqrt{\pi}\cdot \Gamma(n-\frac{1}{2}) }{\Gamma (n)}$$
2026-03-28 10:35:24.1774694124
Expressing $\int_{-\infty}^\infty dx/(x^2+1)^n$ in terms of Gamma function
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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.