Question

How to integrate :
${sin}^{2} x$
​

Answer 1

Answer:

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