Integrate xsin(2x) by dx between the limits 0 and pi/2.

First it is important to identify that this is an integration by parts question as it can't be solved by substitution.
Let I = integral for ease of notation.Write out integration by parts formula I(u)dv= uv -I(v)du. You therefore need to select v and u so that you can integrate by du later on in your analysis.
In this case if we select u = x; du = dx. And if we select dv = sin2x; v = -cos(2x)/2.Then write in form as above I(u)dv = -(xcos(2x))/2 + I(cos(2x) /2 ) dx = -(xcos(2x))/2 +sin(2x)/4
Then sub in the limits to this expression to arrive at an answer of pi/4.