Question

How do you find the integral of `(sin^2(6x))(cos^2(6x))dx`?

Answer 1

You can rewrite this as follows

#int(sin^2(6x))(cos^2(6x))dx=int (1/2*2*sin(6x)*cos(6x))^2dx= int 1/4(sin12x)^2 dx#

Remember that

#cos2x=cos^2x-sin^2x=1-2sin^2x=> sin^2x=1/2*(1-cos2x)#

Hence `1/4int (sin^2 12x)dx=int 1/8(1-cos24x)dx=x/8-1/192*sin(24 x)+c`