Page 45 - Calculus for the Clueless, Calc II

Chapter 8—
Improper Integrals

In discussing an improper integral, it would seem to be a good idea to recall what a proper integral is. In Calc I,
we defined the integral of f(x) from a to b this way: break up the interval (a,b) into n parts. Let w i be any point
in the interval ∆x i. Form the sum f(w 1)∆x 1 + f(w 2)∆x 2 + f(w 3)∆x 3 + ... + f(w n).∆x n. Form the sum . If
the limit exists as n goes to infinity and all the deltas go to zero, we have

At the start, we usually take f(x) to be continuous, although that can be weakened. However, implied in the
definition is that everything is finite; that is, both a and b are finite and f(x) is always finite. What happens if we
have an infinity? In effect we close our eyes and pretend the infinity is not there. We then take the limit as we
go to that infinity. If the limit gives us a single finite number, we will say the integral converges to that number.
Otherwise, the integral diverges. Let us be more formal.

Example 1—

We rewrite this as

You might ask, ''Are they all this easy?" In most books, the vast majority of the improper integrals are relatively
easy in order to make sure that you understand what an improper integral is without worrying about a
complicated integral.

In summary, this integral converges to the value 3π/4.

Example 2—

We write

1/2
But a goes to infinity as . Therefore, this integral diverges.
Note

In the kind of integral of Example 2, if the exponent in the denominator is less than or equal to 1, the integral
diverges. If the exponent is greater than 1, the integral converges.