Quadratic Taylor approximation of an integral

Question

I'm trying to find the quadratic Taylor approximation of $$f(x,y)=\int_0^{x+y^2}e^{-t^2}dt$$about the point $(0,0)$. I really don't know how to solve this though since I don't know even if it's possible to find the partial derivative of an integral. If anyone could provide any advice or hints, that would be great.

Answer 1

Let be $E(s) = \int_0^{s}e^{-t^2}\,dt$ $$f(x,y) = \int_0^{x+y^2}e^{-t^2}\,dt = E(x + y^2).$$ By the chain rule and the fundamental theorem of calculus: $$ \partial_x f(x,y) = E'(x + y^2)\partial_x(x + y^2) = e^{-(x+y^2)^2}. $$ $$\cdots$$