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

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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

$\blacksquare$


Sources