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

## 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

 $\ds \int \sin^2 a x \cos^2 a x \rd x$ $=$ $\ds \int \sin^2 a x \paren {1 - \sin^2 a x} \rd x$ 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 Integrals $\ds$ $=$ $\ds \frac x 2 - \frac {\sin 2 a x} {4 a} - \int \sin^4 a x \rd x + C$ 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

$\blacksquare$

## Proof 2

 $\ds \int \sin^2 a x \cos^2 a x \rd x$ $=$ $\ds \int \paren {\sin a x \cos a x}^2 \rd x$ $\ds$ $=$ $\ds \int \paren {\frac 1 2 \sin 2 a x}^2 \rd x$ 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 $\sin^2 a x$ $\ds$ $=$ $\ds \frac x 8 - \frac {\sin 4 a x} {32 a} + C$

$\blacksquare$