Primitive of Power of Cotangent of a x by Square of Cosecant of a x

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Theorem

$\displaystyle \int \cot^n a x \csc^2 a x \ \mathrm d x = \frac {-\cot^{n + 1} a x} {\left({n + 1}\right) a} + C$


Proof

\(\displaystyle z\) \(=\) \(\displaystyle \cot a x\)
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\mathrm d z} {\mathrm d x}\) \(=\) \(\displaystyle -a \csc^2 a x\) Derivative of Cotangent Function: Corollary
\(\displaystyle \implies \ \ \) \(\displaystyle \int \cot^n a x \csc^2 a x \ \mathrm d x\) \(=\) \(\displaystyle \int \frac {-1} a z^n \ \mathrm d z\) Integration by Substitution
\(\displaystyle \) \(=\) \(\displaystyle \frac {-1} a \frac {z^{n + 1} } {n + 1}\) Primitive of Power
\(\displaystyle \) \(=\) \(\displaystyle \frac {-\cot^{n + 1} a x} {\left({n + 1}\right) a} + C\) substituting for $z$ and simplifying

$\blacksquare$


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