You're ignoring the absolute value signs in . In any interval of length 2π, the function is positive through half the interval and negative through the other half, so its integral over the whole interval is 0. But if you take its absolute value, that will always be positive. So the value of the integral will be twice the value of the integral of over the half of the interval in which it is positive (sketch a graph if this helps you to visualise it).
The function is positive in the interval , so use your expression for the indefinite integral (which by the way is a very good method of doing it), evaluate it between and , and multiply the answer by 2.