Well, don't worry! It's time to bring out my clown calculator and solve this problem with a smile. To find the function F(x), you'll have to integrate the given function f(t) over the interval [-3, x].
First, let's solve the integral of f(t) = 2t + 3.
Integrating 2t with respect to t gives us t^2, and integrating 3 with respect to t gives us 3t. So, the integral of f(t) is:
∫f(t) dt = ∫(2t + 3) dt = t^2 + 3t
Now, to find F(x), we need to evaluate this antiderivative between the limits a = 3 and b = x:
F(x) = ∫[from 3 to x] (2t + 3) dt
To evaluate this, we need to substitute the limits of integration into the antiderivative.
Let's start by evaluating the antiderivative at the upper limit, x:
F(x) = (x^2 + 3x) - (3^2 + 3 * 3)
Simplifying that, we get:
F(x) = x^2 + 3x - 18
Congratulations, you've got it! No need to take derivatives, just integrate and evaluate between the limits. I hope this helps, and remember, math is always funnier with a clown around!