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

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Theorem

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


Proof

\(\displaystyle \) \(\) \(\displaystyle \frac {\sin^{m + 1} a x} {a \left({n - 1}\right) \cos^{n - 1} a x} - \frac {m + 1} {n - 1} \int \frac {\sin^m a x} {\cos^{n - 2} a x} \ \mathrm d x + C\)
\(\displaystyle \) \(=\) \(\displaystyle \int \frac {\sin^{m + 2} a x} {\cos^n a x} \ \mathrm d x\) Primitive of $\dfrac {\sin^m a x} {\cos^n a x}$: Reduction of Both Powers
\(\displaystyle \) \(=\) \(\displaystyle \int \frac {\sin^m a x \left({1 - \cos^2 a x}\right)} {\cos^n a x} \ \mathrm d x\) Sum of Squares of Sine and Cosine
\(\displaystyle \) \(=\) \(\displaystyle \int \frac {\sin^m a x} {\cos^n a x} \ \mathrm d x - \int \frac {\sin^m a x \cos^2 a x} {\cos^n a x} \ \mathrm d x\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \int \frac {\sin^m a x} {\cos^n a x} \ \mathrm d x - \int \frac {\sin^m a x} {\cos^{n - 2} a x} \ \mathrm d x\) simplifying


Hence, rearranging:

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

$\blacksquare$


Also see


Sources