Primitive of Power of Cosine of a x over Power of Sine of a x
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Theorem
Reduction of Power of Cosine
- $\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x = \frac {\cos^{m - 1} a x} {a \paren {m - n} \sin^{n - 1} a x} + \frac {m - 1} {m - n} \int \frac {\cos^{m - 2} a x} {\sin^n a x} \rd x + C$
Reduction of Power of Sine
- $\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x = \frac {-\cos^{m + 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m - n + 2} {n - 1} \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x + C$
Reduction of Both Powers
- $\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x = \frac {-\cos^{m - 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m - 1} {n - 1} \int \frac {\cos^{m - 2} a x} {\sin^{n - 2} a x} \rd x + C$