Primitive of Square of Cosine of a x

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Theorem

$\displaystyle \int \cos^2 a x \, \mathrm d x = \frac x 2 + \frac {\sin 2 a x} {4 a} + C$


Proof

\(\displaystyle \int \cos^2 x \, \mathrm d x\) \(=\) \(\displaystyle \frac x 2 + \frac {\sin 2 x} 4 + C\) Primitive of $\cos^2 x$
\(\displaystyle \implies \ \ \) \(\displaystyle \int \cos^2 a x \, \mathrm d x\) \(=\) \(\displaystyle \frac 1 a \left({\frac {a x} 2 + \frac {\sin 2 a x} 4}\right) + C\) Primitive of Function of Constant Multiple
\(\displaystyle \) \(=\) \(\displaystyle \frac x 2 + \frac {\sin 2 a x} {4 a} + C\) simplifying

$\blacksquare$


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