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

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Theorem

$\ds \int_0^\infty \frac {x^m \rd x} {\paren {a^n + x^n}^r} = \frac {\paren {-1}^{r - 1} \pi a^{m + 1 - n r} \, \map \Gamma {\frac {m + 1} n} } {n \sin \frac {\paren {m + 1} \pi} n \paren {r - 1}! \, \map \Gamma {\frac {m + 1} n - r + 1} }$

for:

$0 < m + 1 < n r$
$r + 1 - \dfrac {m + 1} n \notin \N_{>0}$
$r \in \N_{>0}$.


Proof

\(\ds u\) \(=\) \(\ds \frac {x^n} {a^n}\)
\(\ds \leadsto \ \ \) \(\ds \d u\) \(=\) \(\ds \frac {n x^{n - 1} } {a^n} \rd x\) Power Rule for Derivatives
\(\ds \d u\) \(=\) \(\ds \frac {n \paren {a \times u^{\frac 1 n} }^{n - 1} } {a^n} \rd x\) substituting for $x$
\(\ds \leadsto \ \ \) \(\ds \d x\) \(=\) \(\ds \frac a n u^{\frac 1 n - 1} \rd u\) simplifying
\(\ds \leadsto \ \ \) \(\ds \int_0^\infty \frac {x^m \rd x} {\paren {a^n + x^n}^r}\) \(=\) \(\ds \int_0^\infty \frac {\paren {a \times u^{\frac 1 n} }^m } {\paren {a^n + \paren {a \times u^{\frac 1 n} }^n}^r} \frac a n u^{\frac 1 n - 1} \rd u\) Integration by Substitution
\(\ds \) \(=\) \(\ds \frac {a^{m + 1 - r n} } n \int_0^\infty \frac {\paren {u^{\frac 1 n} }^m} {\paren {1 + \paren {u^{\frac 1 n} }^n}^r} u^{\frac 1 n - 1} \rd u\) factoring out constants
\(\ds \) \(=\) \(\ds \frac {a^{m + 1 - r n} } n \int_0^\infty \frac {u^{\frac {m + 1} n - 1} } {\paren {1 + u}^r} \rd u\) simplifying
\(\ds \) \(=\) \(\ds \frac {a^{m + 1 - r n} } n \map \Beta {\frac {m + 1} n, r - \frac {m + 1} n}\) Beta Function as Integral of Power of $t$ over Power of $t + 1$
\(\ds \) \(=\) \(\ds \frac {a^{m + 1 - r n} } n \frac {\map \Gamma {\frac {m + 1} n} \map \Gamma {r - \frac {m + 1} n} } {\map \Gamma r}\) Definition 3 of Beta Function
\(\ds \) \(=\) \(\ds \frac {\pi a^{m + 1 - r n} \map \Gamma {\frac {m + 1} n} } {n \map \Gamma r} \frac \pi {\map \sin {\pi \paren {r - \frac {m + 1 } n} } \map \Gamma {1 - \paren {r - \frac {m + 1} n} } }\) Euler's Reflection Formula with restriction $r + 1 - \dfrac {m + 1} n \notin \N_{>0}$
\(\ds \) \(=\) \(\ds \frac {\pi a^{m + 1 - r n} \map \Gamma {\frac {m + 1} n} } {n \map \sin {\pi \paren {r - \frac {m + 1} n} } \map \Gamma r \map \Gamma {1 - r + \frac {m + 1} n} }\) merging fractions
\(\ds \) \(=\) \(\ds \frac {\pi a^{m + 1 - r n} \map \Gamma {\frac {m + 1} n} } {n \paren {-1}^r \map \sin {\pi \paren {-\frac {m + 1} n} } \paren {r - 1}! \map \Gamma {1 - r + \frac {m + 1} n} }\) Sine of Angle plus Integer Multiple of Pi and Gamma Function Extends Factorial with restriction $r \in \N_{>0}$
\(\ds \) \(=\) \(\ds \frac {\paren {-1}^{r - 1} \pi a^{m + 1 - r n} \map \Gamma {\frac {m + 1} n} } {n \map \sin {\frac {\paren {m + 1} \pi} n} \paren {r - 1}! \map \Gamma {1 - r + \frac {m + 1} n} }\) simplifying

$\blacksquare$


Sources

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