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

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Theorem

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


Proof

\(\ds \int \frac {\d x} {\sin^m a x \cos^n a x}\) \(=\) \(\ds \int \frac {\cos^{-n} a x \rd x} {\sin^m a x}\)
\(\ds \) \(=\) \(\ds \frac {-\cos^{-n + 1} a x} {a \paren {m - 1} \sin^{m - 1} a x} - \frac {-n - m + 2} {m - 1} \int \frac {\cos^{-n} a x} {\sin^{m - 2} a x} \rd x + C\) Primitive of $\dfrac {\cos^m a x} {\sin^n a x}$
\(\ds \) \(=\) \(\ds \frac {-1} {a \paren {n - 1} \sin^{m - 1} a x \cos^{n - 1} a x} + \frac {m + n - 2} {m - 1} \int \frac {\d x} {\sin^{m - 2} a x \cos^n a x} + C\)

$\blacksquare$


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