Primitive of Reciprocal of Power of Cosine of a x by Power of Sine of a x

From ProofWiki
Jump to navigation Jump to search

Theorem

Reduction of Power of Sine

$\displaystyle \int \frac {\d x} {\sin^m a x \cos^n a x} = \frac {-1} {a \paren {n - 1} \sin^{m - 1} a x \cos^{n - 1} a x} + \frac {m + n - 2} {m - 1} \int \frac {\d x} {\sin^{m - 2} a x \cos^n a x} + C$


Reduction of Power of Cosine

$\displaystyle \int \frac {\mathrm d x} {\sin^m a x \cos^n a x} = \frac 1 {a \left({n - 1}\right) \sin^{m - 1} a x \cos^{n - 1} a x} + \frac {m + n - 2} {n - 1} \int \frac {\mathrm d x} {\sin^m a x \cos^{n - 2} a x}$