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

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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

$\blacksquare$


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