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

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Theorem

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


Proof

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

$\blacksquare$


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