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

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Theorem

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

for $n \ne -1$.


Proof

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

$\blacksquare$


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Sources