Primitive of Power of x over Power of Power of x minus Power of a

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Theorem

$\ds \int \frac {x^m \rd x} {\paren {x^n - a^n}^r} = a^n \int \frac {x^{m - n} \rd x} {\paren {x^n - a^n}^r} + \int \frac {x^{m - n} \rd x} {\paren {x^n - a^n}^{r - 1} }$


Proof

\(\ds \int \frac {x^m \rd x} {\paren {x^n - a^n}^r}\) \(=\) \(\ds \int \frac {x^{m - n} x^n \rd x} {\paren {x^n + a^n}^r}\)
\(\ds \) \(=\) \(\ds \int \frac {x^{m - n} \paren {x^n - a^n + a^n} \rd x} {\paren {x^n - a^n}^r}\)
\(\ds \) \(=\) \(\ds \int \frac {x^{m - n} \paren {x^n - a^n} \rd x} {\paren {x^n - a^n}^r} + a^n \int \frac {x^{m - n} \rd x} {\paren {x^n - a^n}^r}\) Linear Combination of Integrals
\(\ds \) \(=\) \(\ds a^n \int \frac {x^{m - n} \rd x} {\paren {x^n - a^n}^r} + \int \frac {x^{m - n} \rd x} {\paren {x^n - a^n}^{r - 1} }\) simplification

$\blacksquare$


Sources