Question

I have no idea what to do

Answer 1

Don't be scared with the t attached at the end.
This is known as the dummy variable. It's simply an arbitrary variable that represents some value at a particular time.

The next section is the derivative-integral relationship. They are simply inverses of each other; that is, when multiplied together, the two will cancel each other out; much like squares and square roots, as well as exponentials and logarithms.

So, we can simply see the derivative will be sinx + C.
Now, since the limit of the integral are two functions, we can apply the chain rule to them to get our derivative of the integral.


`(d(\int_(f(x))^(g(x)) sint\,dt))/(dx) = f'(x) \cdot sin[f(x)] - g'(x) \cdot sin[g(x)]`

`(d(\int_(x^(3))^(2x) sint\,dt))/(dx) = 3x^(2) \cdot sin[x^(3)] - 2 \cdot sin[2x]`