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

From ProofWiki
Jump to navigation Jump to search

Theorem

$\ds \int \cot^n a x \csc^2 a x \rd x = \frac {-\cot^{n + 1} a x} {\paren {n + 1} a} + C$


Proof

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

$\blacksquare$


Also see


Sources