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

## 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

$\blacksquare$