Primitive of Sine of a x by Cosine of a x

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Theorem

$\displaystyle \int \sin a x \cos a x \rd x = \frac {\sin^2 a x} {2 a} + C$


Proof

\(\displaystyle \int \sin a x \cos a x \rd x\) \(=\) \(\displaystyle \int \frac {\sin 2 a x} 2 \rd x\) Double Angle Formula for Sine
\(\displaystyle \) \(=\) \(\displaystyle \frac {-\cos 2 a x} {4 a} + C\) Primitive of $\cos a x$
\(\displaystyle \) \(=\) \(\displaystyle \frac {-\paren {1 - 2 \sin^2 a x} } {4 a} + C\) Double Angle Formula for Cosine: Corollary 2
\(\displaystyle \) \(=\) \(\displaystyle \frac {-1} {4 a} + \frac {\sin^2 a x} {2 a} + C\) separating fraction
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sin^2 a x} {2 a} + C\) subsuming $\dfrac {-1} {4 a}$ into arbitrary constant

$\blacksquare$


Sources