Question

How do you use the fundamental theorem of calculus to find F'(x) given `F(x)=int (t^2+3t+2)dt` from [-3,x]?

Answer 1

`F'(x) = x^2+3x+2`

Explanation:

If asked to find the derivative of an integral then you should not evaluate the integral but instead use the fundamental theorem of Calculus.

The FTOC tells us that:

`d/dx \ int_a^x \ f(t) \ dt = f(x)` for any constant `a`

(ie the derivative of an integral gives us the original function back).

We are asked to find:

`F'(x)` where `F(x) = int_(-3)^(x) \ t^2+3t+2 \ dt`

ie, we want:

`F'(x) = d/dx int_(-3)^(x) \ t^2+3t+2 \ dt` ..... [A]

And so we can directly apply the FTOC, giving:

`F'(x) = x^2+3x+2`