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

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Theorem

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


Proof

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

$\blacksquare$


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