Example 8

Find

Let u = x - x³. Then du = (1 - 3x²) dx. When x = 0, u = 0 - 0³ = 0. When x = 1, u = 1 - 1³ = 0. Then

As x goes from 0 to 1, u starts at 0, increases for a time, then drops back to 0 (Figure 4.4.7).

Figure 4.4.7

We do not know how to find the indefinite integrals in this example. Nevertheless the answer is 0 because on changing variables both limits of integration become the same. Using the Addition Property, we can also see that, for instance,