Find the integral of sin^2(X)

As soon as you see a question asking you to integrate the square of sin, cos or tan, your first approach should be to use trigonometric identities and double angle formulas.

For sin²(X), we will use the cos double angle formula:
cos(2X) = 1 - 2sin²(X)

The above formula can be rearranged to make sin²(X) the subject:
sin²(X) = 1/2(1 - cos(2X))

You can now rewrite the integration: 
∫sin²(X)dX = ∫1/2(1 - cos(2X))dX

Because 1/2 is a constant, we can remove it from the integration to make the calculation simpler. We are now integrating:
1/2 x ∫(1 - cos(2X)) dX = 1/2 x (X - 1/2sin(2X)) + C

It is very important that as this is not a definite integral, we must add the constant C at the end of the integration.

Simplifying the above equation gives us a final answer:
∫sin²(X) dX = 1/2X - 1/4sin(2X) + C