Primitive of Cube of Sine Function

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Theorem

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


Proof 1

From Primitive of $\sin^3 a x$:

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


The result follows by setting $a = 1$.

$\blacksquare$


Proof 2

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

$\blacksquare$


Sources