Integration of sin²(x) and the half-angle formula

Integration of sin² is easily achieved by first applying the half-angle formula,

`sin^2(x)=(1-cos(2x))/2`

So it is now possible to remove the squared term from the integral,

`int sin^2(x)dx = 1/2int (1-cos(2x))dx`

Exploiting the distributive property of integration, we can expand the integral,

`int sin^2(x)dx = 1/2[int dx - int cos(2x)dx]`

To solve the integral we use u-substitution,

`u=2x`, and so `(du)/(dx)=2`, which we rearrange, `dx=(du)/2`

Therefore,

`int cos(2x)dx = 1/2int cos(u)du = 1/2 sin(u) = 1/2 sin(2x)`

We use this result and substitute it into the integral above,

`int sin^2(x) = 1/2 [x - 1/2sin(2x)]=x/2-1/4sin(2x)`