Primitive of Square of Cosecant of a x

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Theorem

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


Proof

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

$\blacksquare$


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Sources