Evaluate the following integral. ∫ sin 2 x cos 2 x d x

In this solution we will use the double angle formula to help simplify the integral as follows.

$$\begin{array}{cc}\int {\sin }^{2}x{\cos }^{2}xdx& =\int {\left(\sin x\cos x\right)}^{2}dx\\ & =\int {\left(\frac{1}{2}\sin \left(2x\right)\right)}^{2}dx\\ & =\frac{1}{4}\int {\sin }^2\left(2x\right)dx\end{array}$$

Now, we use the half angle formula for sine to reduce to an integral that we can do.

$$\begin{array}{cc}\int {\sin }^{2}x{\cos }^{2}xdx& =\frac{1}{8}\int 1-\cos \left(4x\right)dx\\ & =\frac{1}{8}x-\frac{1}{32}\sin \left(4x\right)+c\end{array}$$