Question

use part 1 of the fundamental theorem of calculus to find the derivative of:

Int. from x to 4 of sin(t^3)dt

Answer 1

you have to start with integrating with respect to t
can you do that?

Answer 2

You want to differentiate with respect to x, right? You don't have to integrate a sin(t^3). Use the idea that differentiation is the inverse of integration.

\[d/dx[\int\limits_{x}^{4}f'(t)dt]=d/dx[f(4)-f(x)]\]

I let f'(t) = sin(t^3). Since sin(t^3) is a continuous function, it must have an antiderivative (we just don't know what it is, nor do we care). We then plug 4 into that antiderivative and subtract f(x) from it.

\[d/dx[f(4)-f(x)]=0-f'(x)*1=-f'(x)\]

Plug x in for t in f'(t) to get your answer: -sin(x^3)