# 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]?

Mar 7, 2018

$F ' \left(x\right) = {x}^{2} + 3 x + 2$

#### 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:

$\frac{d}{\mathrm{dx}} \setminus {\int}_{a}^{x} \setminus f \left(t\right) \setminus \mathrm{dt} = f \left(x\right)$ for any constant $a$

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

$F ' \left(x\right)$ where $F \left(x\right) = {\int}_{- 3}^{x} \setminus {t}^{2} + 3 t + 2 \setminus \mathrm{dt}$
$F ' \left(x\right) = \frac{d}{\mathrm{dx}} {\int}_{- 3}^{x} \setminus {t}^{2} + 3 t + 2 \setminus \mathrm{dt}$ ..... [A]
$F ' \left(x\right) = {x}^{2} + 3 x + 2$