# Primitive of Reciprocal of Power of x by Power of Power of x minus Power of a

## Theorem

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

## Proof

 $\displaystyle \int \frac {\mathrm d x} {x^m \ \left({x^n - a^n}\right)^r}$ $=$ $\displaystyle \int \frac {a^n \ \mathrm d x} {a^n x^m \ \left({x^n - a^n}\right)^r}$ multiplying top and bottom by $a^n$ $\displaystyle$ $=$ $\displaystyle \int \frac {\left({x^n - \left({x^n - a^n}\right)}\right) \ \mathrm d x} {a^n x^m \ \left({x^n + a^n}\right)^r}$ adding and subtracting $x^n$ $\displaystyle$ $=$ $\displaystyle \frac 1 {a^n} \int \frac {x^n \ \mathrm d x} {x^m \ \left({x^n - a^n}\right)^r} - \frac 1 {a^n} \int \frac {\left({x^n - a^n}\right) \ \mathrm d x} {x^m \ \left({x^n - a^n}\right)^r}$ Linear Combination of Integrals $\displaystyle$ $=$ $\displaystyle \frac 1 {a^n} \int \frac {\mathrm d x} {x^{m - n} \left({x^n - a^n}\right)^r} - \frac 1 {a^n} \int \frac {\mathrm d x} {x^m \left({x^n - a^n}\right)^{r - 1} }$ simplification

$\blacksquare$