Primitive of Cube of Sine of a x/Proof 2
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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
| \(\ds \int \sin^3 a x \rd x\) | \(=\) | \(\ds \int \paren {1 - \cos^2 a x} \sin a x \rd x\) | 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 $\sin a x$ |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration: Algebraic Integration: Powers of cos and sine: Example 2.