Primitive of Sine of a x squared by Cosine of a x squared/Proof 2

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

\(\ds \int \sin^2 a x \cos^2 a x \rd x\)

\(=\)

\(\ds \int \paren {\sin a x \cos a x}^2 \rd x\)

\(\ds \)

\(=\)

\(\ds \int \paren {\frac 1 2 \sin 2 a x}^2 \rd x\)

Double Angle Formula for Sine

\(\ds \)

\(=\)

\(\ds \frac 1 4 \int \sin^2 2 a x \rd x\)

\(\ds \)

\(=\)

\(\ds \frac 1 4 \paren {\frac x 2 - \frac {\map \sin {2 x \times 2 a} } {4 \times 2 a} } + C\)

Primitive of $\sin^2 a x$

\(\ds \)

\(=\)

\(\ds \frac x 8 - \frac {\sin 4 a x} {32 a} + C\)

$\blacksquare$