Trying to simplify this integral by using integration by parts. $$ \int\limits_0^1 \int\limits_0^{g(x)} y.f(x,y) dydx $$
I define $$ u=y \space \space and \space \space dv=f(x,y)dy$$ $$ du=dy \space \space and\space \space v= \bf\underline{F(x,y) \space \space or \space v= \space dF(x,y)/dx} $$ What is v above?
Or could you help me about simplifying this integral if you define differently? I guess that there would be probably 2 step integration by parts need to be implemented to simplify completely.
Note: f(x,y) is continuous, differentiable, and integrable.