Let F(x) = ∫ from 1 to x of (t^2 + 4t) dt. Find F'(x).

Step 1: Identify the form of the function.

We are given:
F(x) = ∫ from 1 to x of (t^2 + 4t) dt

The upper limit of integration is the variable x.

Step 2: Recall the Fundamental Theorem of Calculus.

If:
F(x) = ∫ from a to x of f(t) dt

then:
F'(x) = f(x)

In other words, differentiating an accumulation function of this form returns the original integrand evaluated at x.

Step 3: Identify the integrand.

Here:
f(t) = t^2 + 4t

Step 4: Replace t with x.

Therefore:
F'(x) = x^2 + 4x

No integration is necessary because the Fundamental Theorem of Calculus gives the derivative directly.

Therefore, F'(x) = x^2 + 4x.