Primitive of Cube of Sine of a x
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Theorem
- $\ds \int \sin^3 a x \rd x = -\frac {\cos a x} a + \frac {\cos^3 a x} {3 a} + C$
Proof 1
| \(\ds \int \sin^3 a x \rd x\) | \(=\) | \(\ds \int \paren {\frac {3 \sin a x - \sin 3 a x} 4} \rd x\) | Power Reduction Formula for Cube of Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 3 4 \int \sin a x \rd x - \frac 1 4 \int \sin 3 a x \rd x\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 3 4 \paren {\frac {-\cos a x} a} - \frac 1 4 \paren {\frac {-\cos 3 a x} {3 a} } + C\) | Primitive of $\sin a x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {-3 \cos a x} {4 a} + \frac 1 {12 a} \paren {\cos 3 a x} + C\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {-3 \cos a x} {4 a} + \frac 1 {12 a} \paren {4 \cos^3 a x - 3 \cos a x} + C\) | Triple Angle Formula for Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {-3 \cos a x} {4 a} + \frac {\cos^3 a x} {3 a} - \frac {\cos a x} {4 a} + C\) | multipying out | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\frac {\cos a x} a + \frac {\cos^3 a x} {3 a} + C\) | simplifying |
$\blacksquare$
Proof 2
| \(\ds \int \sin^3 a x \rd x\) | \(=\) | \(\ds \int \paren {1 - \cos^2 a x} \sin a x \rd x\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \sin a x \rd x - \int \cos^2 a x \sin a x \rd x\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\dfrac {\cos a x} a - \int \cos^2 a x \sin a x \rd x + C\) | Primitive of $\sin a x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\dfrac {\cos a x} a + \dfrac {\cos^3 a x} a + C\) | Primitive of Power of $\cos a x$ by $\sin a x$ |
$\blacksquare$
Also see
- Primitive of $\cos^3 a x$
- 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 $\sin a x$: $14.349$