Hi, I am in need of some help.

Question

How would I find the derivative of the function:

f(x) = int e^sin(t) dt if the
upper bound is x^2 and the lower bound is cos(x)?

I'm not really sure what to do here since e^sin(t) is not an integratable function. Should I instead integrate the derivative of e^sin(t)?

Answer 1

it would be

e^(sin x^2)*2x - e^(cosx) * (-sinx)

This is just the chain rule via the back door.

If F(t) = ∫f(t) dt
then
∫[u,v] f(t) dt = F(v) - F(u)

Now take the derivative, remembering the chain rule.