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]#?

1 Answer
Feb 17, 2016

Answer:

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.