Primitive of Fourth Power of Cosine of a x

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Theorem

$\displaystyle \int \cos^4 a x \ \mathrm d x = \frac {3 x} 8 + \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} + C$


Proof

\(\displaystyle \int \sin^4 a x \ \mathrm d x\) \(=\) \(\displaystyle \int \left({\frac {3 + 4 \cos 2 x + \cos 4 x} 8}\right) \ \mathrm d x\) Power Reduction Formula for $\cos^4$
\(\displaystyle \) \(=\) \(\displaystyle \frac 3 8 \int \mathrm d x + \frac 1 2 \int \cos 2 a x \ \mathrm d x + \frac 1 8 \int \cos 4 a x \ \mathrm d x\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \frac {3 x} 8 + \frac 1 2 \int \cos 2 a x \ \mathrm d x + \frac 1 8 \int \cos 4 a x \ \mathrm d x + C\) Primitive of Constant
\(\displaystyle \) \(=\) \(\displaystyle \frac {3 x} 8 + \frac 1 2 \left({\frac {\sin 2 a x} {2 a} }\right) + \frac 1 8 \left({\frac {\sin 4 a x} {4 a} }\right) + C\) Primitive of $\cos a x$
\(\displaystyle \) \(=\) \(\displaystyle \frac {3 x} 8 + \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} + C\) simplifying

$\blacksquare$


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