Primitive of Cube of Cosine of a x

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Theorem

$\displaystyle \int \cos^3 a x \rd x = \frac {\sin a x} a - \frac {\sin^3 a x} {3 a} + C$


Proof

\(\displaystyle \int \cos^3 a x \rd x\) \(=\) \(\displaystyle \int \left({\frac {3 \cos a x + \cos 3 a x} 4}\right) \rd x\) Power Reduction Formula for $\cos^3$
\(\displaystyle \) \(=\) \(\displaystyle \frac 3 4 \int \cos a x \rd x + \frac 1 4 \int \cos 3 a x \rd x\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \frac 3 4 \left({\frac {\sin a x} a}\right) + \frac 1 4 \left({\frac {\sin 3 a x} {3 a} }\right) + C\) Primitive of $\cos a x$
\(\displaystyle \) \(=\) \(\displaystyle \frac {3 \sin a x} {4 a} + \frac 1 {12 a} \left({\sin 3 a x}\right) + C\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \frac {3 \sin a x} {4 a} + \frac 1 {12 a} \left({3 \sin a x - 4 \sin^3 a x}\right) + C\) Triple Angle Formula for Sine
\(\displaystyle \) \(=\) \(\displaystyle \frac {3 \sin a x} {4 a} + \frac {\sin a x} {4 a} - \frac {\sin^3 a x} {3 a} + C\) multipying out
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sin a x} a - \frac {\sin^3 a x} {3 a} + C\) simplifying

$\blacksquare$


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