Primitive of Sine of a x squared by Cosine of a x squared/Proof 1
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Theorem
- $\ds \int \sin^2 a x \cos^2 a x \rd x = \frac x 8 - \frac {\sin 4 a x} {32 a} + C$
Proof
\(\ds \int \sin^2 a x \cos^2 a x \rd x\) | \(=\) | \(\ds \int \sin^2 a x \paren {1 - \sin^2 a x} \rd x\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \sin^2 a x \rd x - \int \sin^4 a x \rd x\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac x 2 - \frac {\sin 2 a x} {4 a} - \int \sin^4 a x \rd x + C\) | Primitive of $\sin^2 a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac x 2 - \frac {\sin 2 a x} {4 a} - \paren {\frac {3 x} 8 - \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} } + C\) | Primitive of $\sin^4 a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac x 8 - \frac {\sin 4 a x} {32 a} + C\) | gathering terms and simplifying |
$\blacksquare$