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6.5. IMPROPER INTEGRALS
(c)
4
1
1
√
4−x
dx
(d)
2
−2
1
x 2 dx
(e)
π/2
0
tan(x) dx
(f)
1
0
1
√
1−x 2 dx
⊳
Summary
In this section, we encountered the following important ideas:
• An integral
b
a
f (x) dx can be improper if at least one of a or b is ±∞, making the
interval unbounded, or if f has a vertical asymptote at x = c for some value of c that
satisfies a ≤ c ≤ b. One reason that improper integrals are important is that certain
probabilities can be represented by integrals that involve infinite limits.
• When we encounter an improper integral, we work to understand it by replacing the
improper integral with a limit of proper integrals. For instance, we write
∞
a
f (x) dx = lim
b→∞
b
a
f (x) dx,
and then work to determine whether the limit exists and is finite. For any improper
integral, if the resulting limit of proper integrals exists and is finite, we say the improper
integral converges. Otherwise, the improper integral diverges.
• An important class of improper integrals is given by
∞
1
1
x p dx
where p is a positive real number. We can show that this improper integral converges
whenever p > 1, and diverges whenever 0 < p ≤ 1. A related class of improper integrals
is
1
0
1
x p dx, which converges for 0 < p < 1, and diverges for p ≥ 1.
Exercises
1. Determine, with justification, whether each of the following improper integrals converges
or diverges.
(a)
∞
e
ln(x)
x
dx