Alright, I want to see if I understand the language of these two problems and their solutions. It asks:

If F(x) = [given integrand], find the derivative F'(x).
So is F(x) just our function, and F'(x) our antiderivative?

1) F(x) = ∫(2, x) [t^(3)+1]^(1/2)dt
Let u = [t^(3)+1]
∫(2, x) (u)^(1/2)du, all terms in u so
F'(x) = ∫(2, x) 2u^(3/2)/3
F'(x) = ∫(2, x) [2(t^(3)+1)^(3/2)]/3 dt

2) F(x) = ∫(2, x) sin(t^5)dt
FTCI right? We know the derivative of sin as an integral, so:
F'(x) = ∫(2, x) -cos(t^5)dt

We just leave the answers like that with dt? No need for C as our constant? I should not mix up ∫(2, x) as an indefinite ∫, right?

I am not sure I follow you but

when you change from t to u
you must also change dt to du
{ like if u = t^3 + 1, then du = 3 t^2 dt
and dt = du/3 t^2
which leaves you in a mess.

You must also change the t limits, 2 to x
to u limits
if t = 2, u = 2^3+1 = 9
and if t = x then u = x^3+1

You have a general understanding of the language used in these two problems, but there are a few points that I can clarify:

1) In the problem where you are asked to find the derivative, F(x) represents the antiderivative of the given integrand. F'(x) represents the derivative of F(x), which is the original function.

For the first problem:
- You correctly set u = t^3 + 1.
- Then, when rewriting the given integral in terms of u, you should also change the limits of integration. Since u is a function of t, the lower limit of integration becomes u(2) and the upper limit becomes u(x).
- Thus, the derivative F'(x) should be:
F'(x) = ∫(u(2), u(x)) (2u^(3/2))/3 du
= ∫(u(2), u(x)) (2(t^3 + 1)^(3/2))/3 dt

2) For the second problem, you are correct in using the Fundamental Theorem of Calculus. The derivative of F(x) is simply the original function inside the integral, but with the new variable x as the upper limit and the original lower limit. Therefore, the derivative is:
F'(x) = -cos(x^5)

When writing the answers, you should replace the "dt" or "du" notation with the appropriate differential variable, which in this case is "dt". Additionally, since we are dealing with definite integrals (with specified limits of integration), you should not include the constant of integration "C". The definite integral already takes care of accounting for any constant term.

So, the final answers for both problems would be written as:
1) F'(x) = ∫(2, x) [2(t^(3)+1)^(3/2)]/3 dt
2) F'(x) = ∫(2, x) -cos(t^5)dt