Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Power of Sine

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Theorem

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


Proof

\(\ds \) \(\) \(\ds \frac {-\cos^{m + 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m + 1} {n - 1} \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x + C\)
\(\ds \) \(=\) \(\ds \int \frac {\cos^{m + 2} a x} {\sin^n a x} \rd x\) Primitive of $\dfrac {\cos^m a x} {\sin^n a x}$: Reduction of Both Powers
\(\ds \) \(=\) \(\ds \int \frac {\cos^m a x \paren {1 - \sin^2 a x} } {\sin^n a x} \rd x\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x - \int \frac {\cos^m a x \sin^2 a x} {\sin^n a x} \rd x\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x - \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x\) simplifying


Hence, rearranging:

\(\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x\) \(=\) \(\ds \frac {-\cos^{m + 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m + 1} {n - 1} \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x + \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x + C\)
\(\ds \) \(=\) \(\ds \frac {-\cos^{m + 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m + 1} {n - 1} \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x - \frac {-n + 1} {n - 1} \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x + C\) common denominator
\(\ds \) \(=\) \(\ds \frac {-\cos^{m + 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m - n + 2} {n - 1} \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x + C\)

$\blacksquare$


Also see


Sources