How to calculate a double trig integral with limits as x and y?

Question

How to go from $$u(x,y) =\sin x+\sin y+\int_0^x d\xi \int_0^y \left( \cos(\xi\eta) -\xi \eta \sin(\xi\eta) \right) d\eta$$ to $$u(x,y) = \sin(x) +\sin y+\sin(xy) $$ in full? This is an example for the solution to the Goursat problem.

Answer 1

The $\eta$ integral gives $y \cos {\xi y}$, then the $\xi$ integral gives $\sin(xy)$ doing it by parts with $y$ and $\cos(\xi y)d\xi$