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
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