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]?

1 Answer
Mar 7, 2018

Answer:

# 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 #