Primitive of Cube of Cosine of a x
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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
- Primitive of $\sin^3 a x$
- Primitive of $\tan^3 a x$
- Primitive of $\cot^3 a x$
- Primitive of $\sec^3 a x$
- Primitive of $\csc^3 a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cos a x$: $14.379$