Problem

What is

Simplify your answer as much as possible using the Fundamental Theorem of Calculus.

Solution

We'll apply the product rule, the Fundamental Theorem of Calculus, and the chain rule.

First, use the product rule, resulting in

Now the Fundamental Theorem of Calculus. Rewrite the sum in a form where the lower bounds are constants.

Assume F(t) is an antiderivative of

Then the above sum equals

Multiply through and simplify to get

The last term in this sum includes a factor that is the derivative of a constant, thus it equals zero. We use the chain rule to evaluate the third term:

Part II of the Fundamental Theorem of Calculus allows us to simplify the first two terms in the sum if we know an antiderivative of f(x). However, at this point we do not have all the techniques we need to find the antiderivative, so we leave the expression in this hybrid form: