Gud question .... But always keep in mind that when you are using integration by parts, you will have to break the original function in such a way that one of them is conveniently integrable and the other differentiable.
In this case : 2sin(x) is integrable as well as differentiable but , 1/3+sin(2x) cannot be integrated ....
So , we break the function and integrate 2sin(x) and differentiate 1/3 + sin(2x) and proceed...
Does this answer your question ???
Always remember ... when a question of integration is asked in IIT-JEE / AIEEE or any other engineering examination ... There is a solution ... It is just about thinking in such a way that you get there easily and quickly !!!!