Primitive of Square of Cosecant of a x over Cotangent of a x

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Theorem

$\displaystyle \int \frac {\csc^2 a x \ \mathrm d x} {\cot a x} = \frac {-\ln \left\vert{\cot a x}\right\vert} a + C$


Proof

\(\displaystyle \frac {\mathrm d} {\mathrm d x} \cot x\) \(=\) \(\displaystyle -\csc^2 x\) Derivative of Cotangent Function
\(\displaystyle \implies \ \ \) \(\displaystyle \int \frac {\csc^2 x \ \mathrm d x} {\cot x}\) \(=\) \(\displaystyle -\ln \left\vert{\cot a x}\right\vert + C\) Primitive of Function under its Derivative
\(\displaystyle \implies \ \ \) \(\displaystyle \int \frac {\csc^2 a x \ \mathrm d x} {\cot a x}\) \(=\) \(\displaystyle \frac 1 a \left({-\ln \left\vert{\cot a x}\right\vert}\right) + C\) Primitive of Function of Constant Multiple
\(\displaystyle \) \(=\) \(\displaystyle \frac {-\ln \left\vert{\cot a x}\right\vert} a + C\) simplifying

$\blacksquare$


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