It should be obvious that the sum of sin(2x) and -sin(2x) is 0 and so the sum of the integrals will be 0!
The basic definition of "absolute value" is that |x|= x if , -x is x< 0.
So |sin(2x)|= sin(2x) if and -sin(2x) if which is the same as
|sin(2x)|= sin(2x) if [tex]0\le x\le \pi/2[/itex] and -sin(2x) if
If you want to integrate from 0 to , you will also need to use the fact that, sine is periodic with period , sin(x) is positive for and negative for so |sin(2x)|= sin(2x) for and |sin(2x)|= -sin(2x) for