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

## Theorem

$\displaystyle \int \frac {x^{n - 1} \rd x} {x^n - a^n} = \frac 1 n \ln \size {x^n - a^n} + C$

## Proof

 $\displaystyle u$ $=$ $\displaystyle x^n - a^n$ $\displaystyle \leadsto \ \$ $\displaystyle \frac {\d u} {\d x}$ $=$ $\displaystyle n x^{n - 1}$ Power Rule for Derivatives and Derivative of Constant $\displaystyle \leadsto \ \$ $\displaystyle \int \frac {x^{n - 1} \rd x} {x^n - a^n}$ $=$ $\displaystyle \frac 1 n \ln \size {x^n - a^n} + C$ Primitive of Function under its Derivative

$\blacksquare$