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

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Theorem

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

for $n \ne -1$.


Proof

\(\ds z\) \(=\) \(\ds \cos a x\) Werner Formula for Sine by Cosine
\(\ds \leadsto \ \ \) \(\ds \frac {\d z} {\d x}\) \(=\) \(\ds -a \sin a x\) Primitive of $\cos a x$
\(\ds \leadsto \ \ \) \(\ds \int \cos^n a x \sin a x \rd x\) \(=\) \(\ds \int \frac {-z^n \rd x} a\) Integration by Substitution
\(\ds \) \(=\) \(\ds \frac {-z^{n + 1} } {\paren {n + 1} a} + C\) Primitive of Power
\(\ds \) \(=\) \(\ds \frac {-\cos^{n + 1} a x} {\paren {n + 1} a} + C\) substituting for $z$

$\blacksquare$


Also see

For $n = -1$, use Primitive of $\dfrac {\sin a x} {\cos a x}$.


Sources