Primitive of Power of Sine of a x by Power of Cosine of a x

Theorem

Reduction of Power of Sine

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

for $n \ne -m$.

Reduction of Power of Cosine

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

for $n \ne -m$.

Examples

Primitive of $\cos^3 x \sin^4 x$

$\ds \int \cos^3 x \sin^4 x \rd x = \dfrac {\sin^5 x} 5 - \dfrac {\sin^7 x} 7 + C$

Primitive of $\sin^2 x \cos^3 x$

$\ds \int \sin^2 x \cos^3 x \rd x = \dfrac {\sin^3 x} 3 - \dfrac {\sin^5 x} 5 + C$

Primitive of $\cos^2 x \sin^4 x$

$\ds \int \cos^2 x \sin^4 x \rd x = \dfrac 1 {32} \paren {2 x - \dfrac {\sin 2 x} 2 - \dfrac {2 \sin 4 x} 4 + \dfrac {\sin 6 x} 6} + C$

Also see

• Primitive of $\tan^n a x$ for $n = -m$.
• Primitive of $\cot^n a x$ for $m = -n$.