Question

How do you use the second fundamental theorem of Calculus to find the derivative of given `int (t^2 +3t+2)dt` from `[-3, x]`?

Answer 1

The second Fundamental Theorem of Calculus enables us to find the function defined by `g(x) = int_-3^x (t^2 +3t+2)dt`. We will then differentiate.

Explanation:

Find an antiderivative of `t^2 +3t+2` and evaluate from `-3` to `x`

`g(x) = int_-3^x (t^2 +3t+2)dt = ]_-3^x`

`= ((x)^3/3+(3(x)^2)/2+2(x))-((-3)^3/3+(3(-3)^2)/2+2(-3))`

`= x^3/3+(3x^2)/2+2x+3/2`

So, `g'(x) = x^2+3x+2`

Note
We get the same answer much more quickly by using part 1 of the fundamental theorem.