Now remember that since you are taking the integral with respect to , you keep constant.
Do you see how instead of having a CONSTANT of integration, we have a FUNCTION of integration? This is because if you partially differentiate a function of with respect to , it becomes .
You also have
This is quite strange. Usually when you do the integrations, you get the same functions, only differing by a function of and a function of . Of course, by putting them together you get the entire function. Are you sure you copied down the question correctly?