Problem

What is

Solution

This looks like a nasty function to take the antiderivative of, but we will first rewrite the integral as the sum of two integrals, using Property 3 and Definition 1 in section 4.3:

We put the integral in this form so that the lower bounds are constants. Then the integrals look like the one in Part 1 of the Fundamental Theorem of Calculus.

Assume F(t) is an antiderivative of

The above expression becomes

The F(0) terms cancel. The derivative of –F(x) gives –f(x), and the derivative of F(x⁸) gives

where we use the chain rule since we substituted x⁸ for x. Therefore the solution is