Definite Integral from 0 to Infinity of x^m over (x^n + a^n)^r

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Theorem

$\displaystyle \int_0^a \frac {x^m \rd x} {\left({a^n - x^n}\right)^r} = \frac {\left({-1}\right)^{r - 1} \pi a^{m + 1 + n r} \, \Gamma \left({\frac {m + 1} n}\right) } {n \sin \frac {\left({m + 1}\right) \pi} n \left({r - 1}\right)! \Gamma \left({\frac {m + 1} n - r + 1}\right)}$

for $0 < m + 1 < n r$.

Proof


Sources