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
- $\ds \int \frac {\d x} {\sin^m a x \cos^n a x} = \frac 1 {a \paren {n - 1} \sin^{m - 1} a x \cos^{n - 1} a x} + \frac {m + n - 2} {n - 1} \int \frac {\d x} {\sin^m a x \cos^{n - 2} a x}$
Proof
| \(\ds \int \frac {\d x} {\sin^m a x \cos^n a x}\) | \(=\) | \(\ds \int \frac {\sin^{-m} a x \rd x} {\cos^n a x}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sin^{-m + 1} a x} {a \paren {n - 1} \cos^{n - 1} a x} - \frac {-m - n + 2} {n - 1} \int \frac {\sin^{-m} a x} {\cos^{n - 2} a x} \rd x + C\) | Primitive of $\dfrac {\sin^m a x} {\cos^n a x}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {a \paren {n - 1} \sin^{m - 1} a x \cos^{n - 1} a x} + \frac {m + n - 2} {n - 1} \int \frac {\d x} {\sin^m a x \cos^{n - 2} a x} + C\) |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.428$