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.
[tex]\frac{d(\int_{f(x)}^{g(x)} sint\,dt)}{dx} = f'(x) \cdot sin[f(x)] - g'(x) \cdot sin[g(x)][/tex]
[tex]\frac{d(\int_{x^{3}}^{2x} sint\,dt)}{dx} = 3x^{2} \cdot sin[x^{3}] - 2 \cdot sin[2x][/tex]
