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

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Theorem

$\displaystyle \int \frac {x^m \ \mathrm d x} {\left({x^n - a^n}\right)^r} = a^n \int \frac {x^{m - n} \ \mathrm d x} {\left({x^n - a^n}\right)^r} + \int \frac {x^{m - n} \ \mathrm d x} {\left({x^n - a^n}\right)^{r - 1} }$


Proof

\(\displaystyle \int \frac {x^m \ \mathrm d x} {\left({x^n - a^n}\right)^r}\) \(=\) \(\displaystyle \int \frac {x^{m - n} x^n \ \mathrm d x} {\left({x^n + a^n}\right)^r}\)
\(\displaystyle \) \(=\) \(\displaystyle \int \frac {x^{m - n} \left({x^n - a^n + a^n}\right) \ \mathrm d x} {\left({x^n - a^n}\right)^r}\)
\(\displaystyle \) \(=\) \(\displaystyle \int \frac {x^{m - n} \left({x^n - a^n}\right) \ \mathrm d x} {\left({x^n - a^n}\right)^r} + a_n \int \frac {x^{m - n} \ \mathrm d x} {\left({x^n - a^n}\right)^r}\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle a^n \int \frac {x^{m - n} \ \mathrm d x} {\left({x^n - a^n}\right)^r} + \int \frac {x^{m - n} \ \mathrm d x} {\left({x^n - a^n}\right)^{r - 1} }\) simplification

$\blacksquare$


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