Example 10

Find

.

We see that

and

Thus diverges and has the form ∞ - ∞. We do not assign it any value or either of the symbols ∞ or -∞. The region under the curve f(x) = x is shown in Figure 6.7.10.

Figure 6.7.10

It is tempting to argue that the positive area to the right of the origin and the negative area to the left exactly cancel each other out so that the improper integral is zero. But this leads to a paradox.

Wrong:

Let

v = x + 2, dv = dx.

Then

Subtracting

But

So we do not give the integral the value 0, and instead leave it undefined.