Integral of (cos(x))^2 or (sin(x))^2

When we try to integrate a function of x, we often use reverse chain rule looking for the derivative of our functions with the power on the side of our integral. However, with these two we do not have an extra sin(x) or cos(x0 respectively on the side and so we cannot integrate this how we normally would.The key here is to be aware of your trigonometric double angle identities.We know cos(2x)= (cos(x))^2 - (sin(x))^2, so if we want to integrate (sin(x))^2 for example, we sub in 1-(sin(x))^2 for our (cos(x))^2 in this double angle identity and then rearrange for (sin(x))^2 and now we will be integrating 0.5(1-cos(2x)) which is now a very standard trigonometric integral and the same can be done for our cos example.