When calculating

we need to remember that

which allows us to turn the ${\cos }^{2}x$ into something we can integrate.

Something very similar works for integrating ${\sin }^{2}x$, as we'll see in the examples.

Find the following integrals:

1. $$\int {\cos }^2(x)dx$$
2. $$\int {\sin }^2(x)dx$$
3. $$\int {\sin }^2(2x)-{\cos }^2\left(\frac{x}{3}\right)dx$$