While studying for an exam I came across an old exam question;
Consider the function
$ F(x) = x^2 \int_{x^3}^1{e^{t^2} dt} $
and compute $ dF/dx $.
As far as I know, this is an unsolvable integral but since the question is asking for the derivative, I figured you wouldn't need to determine the integral at all. The problem I'm having is the factor $ x^2 $.
I know how to solve this if the factor $ x^2 $ was not in there, you swap the lower and upper limit and change the sign in front of the integral, then you just apply the fundamental theorem of calculus (with the chain rule) and get $ dF/dx = -3x^2 * e^{x^6} $. However, with the factor $ x^2 $ present, I have no idea how to solve it. It feels like you would need to use the product rule, which in this case would have the disastrous consequence of having to determine the integral. Is there any smart tricks to solving this?