Electromagnetism Basics
That integral cumulates the place difference between the endpoint and the startpoint. The total transition vector is the X unit vector multiplied by -2 times the radius of the circle. The problem does not search a formal calculation but the most basic understanding of the vector integration along a path.

Your formal calculation which splits the integral to three parts is not ok as you suspected, because the unit vectors are not constant. You must express the unit vectors as rotated X and Y unit vectors. Those are constant, only sin and cos terms must be left inside the integrals. 

The mid part from (0,1) to (0,-1) does not need rotation.