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

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Theorem

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


Proof

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

$\blacksquare$


Sources