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

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Theorem

$\ds \int \csc^n a x \cot a x \rd x = \frac {-\csc^n a x} {n a} + C$

for $n \ne 0$.


Proof

\(\ds z\) \(=\) \(\ds \csc a x\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d z} {\d x}\) \(=\) \(\ds -a \csc a x \cot a x\) Derivative of Cosecant Function, Chain Rule for Derivatives
\(\ds \leadsto \ \ \) \(\ds \int \csc^n a x \cot a x \rd x\) \(=\) \(\ds \int \frac {-z^{n - 1} \rd z} a\) Integration by Substitution
\(\ds \) \(=\) \(\ds \frac {-z^n} {n a}\) Primitive of Power
\(\ds \) \(=\) \(\ds \frac {-\csc^n a x} {n a} + C\) substituting for $z$

$\blacksquare$


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