What is the integral of sin^2(ax)cos^2(ax) ?

The integral of sin^2(ax)cos^2(ax) is 1/8 (x-1/(4a)sin(4xa))+C

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integral of sin^2(ax)cos^2(ax)

\int sin^{2} (a x) cos^{2} (a x) d x

\frac{1}{8} (x - \frac{1}{4 a} sin (4 x a)) + C

Solution steps

\int sin^{2} (a x) cos^{2} (a x) d x

Rewrite using trig identities

= \int \frac{1 - cos ( 4 x a )}{8} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{8} \cdot \int 1 - cos (4 x a) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{8} (\int 1 d x - \int cos (4 x a) d x)

\int 1 d x = x

\int cos (4 x a) d x = \frac{1}{4 a} sin (4 x a)

= \frac{1}{8} (x - \frac{1}{4 a} sin (4 x a))

Add a constant to the solution

= \frac{1}{8} (x - \frac{1}{4 a} sin (4 x a)) + C

What is the integral of sin^2(ax)cos^2(ax) ?
The integral of sin^2(ax)cos^2(ax) is 1/8 (x-1/(4a)sin(4xa))+C