<s>[INST] Prove $\int_{-\infty}^{\infty} \frac{du}{(1+u^{2})^{\frac{7-p}{2}}} =\frac{\sqrt{\pi}\Gamma[\frac{1}{2}(6-p)]}{\Gamma[\frac{1}{2}(7-p)]}$ I try to evaluate following integral
$\int_{-\infty}^{\infty} \frac{du}{(1+u^{2})^{\frac{7-p}{2}}} =\frac{\sqrt{\pi}\Gamma[\frac{1}{2}(6-p)]}{\Gamma[\frac{1}{2}(7-p)]}$
It seems okay to extend
$\int_{-\infty}^{\infty} \frac{du}{(1+u^{2})^{\frac{p}{2}}} =\frac{\sqrt{\pi}\Gamma[\frac{1}{2}(p-1)]}{\Gamma[\frac{p}{2}]}$
It seems it is also related with Beta function, but i could not find how to do. 
 [/INST] The integrand is even, so we can change the limits to $0 \le u \le \infty$ and double the answer.
$I = 2\displaystyle\int\limits_0^{\infty} \dfrac{1}{(1 + u^2)^{(7 - p)/2}} du$
Let $u = \tan x$, then $du = \sec^2 x \,dx$.
$\begin{align}
\therefore I & = 2\int_0^{\pi/2}\dfrac{1}{(\sec x)^{(7 - p)}}\sec^2 x \,dx\\
& = 2\int_0^{\pi/2} (\sec x)^{p - 5} \,dx\\
& = 2\int_0^{\pi/2} \sin^0 x \,\cos^{5 - p} x \,dx\\
& = \beta\left(\dfrac 1 2, \dfrac{6 - p}{2} \right)\\
& = \dfrac{\Gamma\left( \dfrac 1 2 \right) \Gamma \left( \dfrac{6 - p}{2} \right)}{\Gamma\left(\dfrac{7 - p}{2} \right)}\\
& = \boxed{\sqrt{\pi} \dfrac{\Gamma\left[ \frac 1 2 (6 - p) \right]}{\Gamma\left[ \frac 1 2 (7 - p) \right]}}
\end{align}$

Note:

$\displaystyle\int\limits_0^{\pi/2} \cos^p \theta \sin^q \theta \,d\theta = \beta\left( \dfrac{p + 1}{2}, \dfrac{q + 1}{2} \right)$
$\beta(m, n) = \dfrac{\Gamma(m)\Gamma(n)}{\Gamma(m + n)}$

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