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

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Theorem

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


Proof

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

$\blacksquare$


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