Section 7.4: Improper Integrals
Recall that the definite integral of a continuous function f (x) on a closed, bounded
interval [a, b] is defined by
Z b
f (x)dx.
a
In this section, we extend the concept of definite integrals to infinite intervals and discontinuous functions. There are two types of improper integrals.
Type I: (Infinite Intervals)
Integrals where one (or both) of the limits of integration is infinite.
1. If f (x) is continuous on [a, ∞), then
∞
Z
Z
N
f (x)dx.
f (x)dx = lim
N →∞
a
a
2. If f (x) is continuous on (−∞, b], then
Z
b
Z
b
f (x)dx = lim
N →−∞
−∞
f (x)dx.
N
3. If f (x) is continuous on (−∞, ∞), then
Z ∞
Z a
Z
f (x)dx =
f (x)dx +
−∞
−∞
∞
f (x)dx.
a
Definition: Improper integrals are called convergent if the limit exists and is finite. If the
limit is infinite or does not exist, the improper integral is called divergent.
Z
Example: Evaluate
1
∞
1
√ dx.
x
1
Z
∞
Example: Evaluate
1
1
dx.
x2
Note: The improper integral
Z
1
∞
1
dx
xp
was convergent for p = 2 and divergent for p = 1/2. For which values of p does this improper
integral converge?
Z
Theorem: The improper integral
a
Z
Example: Evaluate
e
∞
∞
1
dx converges for p > 1 and diverges for p ≤ 1.
xp
1
dx.
x ln x
2
Z
0
cos(2x)dx.
Example: Evaluate
−∞
Z
∞
Example: Evaluate
−∞
dx
dx.
+1
x2
3
Type II: (Discontinuous Integrands)
Integrals where f (x) has a discontinuity on [a, b].
1. If f (x) is continuous on [a, b) and discontinuous at x = b, then
b
Z
f (x)dx = lim−
N →b
a
N
Z
f (x)dx.
a
2. If f (x) is continuous on (a, b] and discontinuous at x = a, then
Z
a
b
Z
f (x)dx = lim+
N →a
b
f (x)dx.
N
3. If f (x) is discontinuous at x = c for a < c < b, then
Z
b
c
Z
f (x)dx =
a
Z
f (x)dx +
a
Example: Evaluate the following definite integrals.
Z 3
1
√ dx
(a)
x
0
Z
(b)
1
9
1
√
dx
3
x−9
4
b
f (x)dx.
c
Z
1
1
dx
x2
(c)
−1
Theorem: (Comparison Theorem for Improper Integrals)
Suppose f and g are continuous functions with f (x) ≥ g(x) ≥ 0 on [a, ∞).
Z ∞
Z ∞
g(x)dx.
f (x)dx converges, then so does
1. If
a
a
Z
2. If
∞
Z
∞
g(x)dx diverges, then so does
a
f (x)dx.
a
Example: Determine whether the following integrals converge or diverge.
Z ∞
x
dx
(a)
4
x +2
1
5
Z
∞
1 + e−x
dx
x
∞
dx
x + e2x
(b)
2
Z
(c)
2
6