Primitive of Sine of a x squared by Cosine of a x squared

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Theorem

$\displaystyle \int \sin^2 a x \cos^2 a x \rd x = \frac x 8 - \frac {\sin 4 a x} {32 a} + C$


Proof 1

\(\displaystyle \int \sin^2 a x \cos^2 a x \rd x\) \(=\) \(\displaystyle \int \sin^2 a x \paren {1 - \sin^2 a x} \rd x\) Sum of Squares of Sine and Cosine
\(\displaystyle \) \(=\) \(\displaystyle \int \sin^2 a x \rd x - \int \sin^4 a x \rd x\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \frac x 2 - \frac {\sin 2 a x} {4 a} - \int \sin^4 a x \rd x + C\) Primitive of $\sin^2 a x$
\(\displaystyle \) \(=\) \(\displaystyle \frac x 2 - \frac {\sin 2 a x} {4 a} - \paren {\frac {3 x} 8 - \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} } + C\) Primitive of $\sin^4 a x$
\(\displaystyle \) \(=\) \(\displaystyle \frac x 8 - \frac {\sin 4 a x} {32 a} + C\) gathering terms and simplifying

$\blacksquare$


Proof 2

\(\displaystyle \int \sin^2 a x \cos^2 a x \rd x\) \(=\) \(\displaystyle \int \paren {\sin a x \cos a x}^2 \rd x\)
\(\displaystyle \) \(=\) \(\displaystyle \int \paren {\frac 1 2 \sin 2 a x}^2 \rd x\) Double Angle Formula for Sine
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 4 \int \sin^2 2 a x \rd x\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 4 \paren {\frac x 2 - \frac {\map \sin {2 x \times 2 a} } {4 \times 2 a} } + C\) Primitive of $\sin^2 a x$
\(\displaystyle \) \(=\) \(\displaystyle \frac x 8 - \frac {\sin 4 a x} {32 a} + C\)

$\blacksquare$


Sources