Primitive of Fourth Power of Cosine of a x

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Theorem

$\ds \int \cos^4 a x \rd x = \frac {3 x} 8 + \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} + C$


Proof

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

$\blacksquare$


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