Primitive of Cube of Cosine of a x

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Proof

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

$\blacksquare$


Also see


Sources