Integration using Double Angle Formula

$$ \begin{align} \displaystyle \cos{2x} &= 2\cos^2{x} - 1 \\ &= 1-2 \sin^2 {x} \\ &= \cos^2{x} - \sin^2{x} \\ \sin{2x} &= 2 \sin{x} \cos{x} \\ \end{align} $$

Example 1

Find $\displaystyle \int{\sin^2{x}}dx$.

Example 2

Find $\displaystyle \int{\cos^2{x}}dx$.

Example 3

Find $\displaystyle \int{\sin^2{x}}dx$.

Example 4

Find $\displaystyle \int{\cos^2{6x}}dx$.