Chapter 8. Techniques of integration
Section 8.9 Improper integrals
In this section we extend the conception of a definite integral to the case where the interval
is infinite and also to the case where integrand is unbounded.
Definition of an improper integral of type 1 (infinite intervals)
(a) If
Rt
f (x)dx exists for every number t ≥ a, then
a
Z∞
Zt
f (x)dx = lim
f (x)dx
t→∞
a
a
provided this limit exists (as a finite number)
Rb
(b) If f (x)dx exists for every number t ≤ b, then
t
Zb
Zb
f (x)dx = lim
f (x)dx
t→−∞
−∞
t
provided this limit exists (as a finite number)
The improper integrals in (a) and (b) are called convergent if the limit exist and divergent
if the limit does not exist.
R∞
Rb
(c) If both f (x)dx and
f (x)dx are convergent, then we define
a
−∞
Z∞
f (x)dx =
−∞
Z∞
Za
f (x)dx +
−∞
f (x)dx
a
where a is any real number
Example 1. For what values of p is the integral
1
R∞ 1
dx convergent?
p
1 x
Example 2. Determine whether each integral is convergent or divergent. Evaluate those
that are convergent.
Z∞
1.
√
dx
x+3
2
Z3
dx
x2 + 9
2.
−∞
Z−1
3.
dx
√
3
x x−1
−∞
4.
R∞
(2x2 + x − 1)dx
−∞
2
Z∞
5.
1
dx
(x + 2)(x + 3)
0
6.
R∞
ex dx
1
7.
R1
ex dx
−∞
8.
R∞
ex dx
−∞
3
Z∞
9.
dx
x(ln x)2
e
Definition of an improper integral of type 2 (discontinuous integrands)
(a) If f is continuous on [a, b) and is discontinuous at b, then
Zb
Zt
f (x)dx = lim−
f (x)dx
t→b
a
a
if this limit exists (as a finite number)
(b) If f is continuous on (a, b] and is discontinuous at a, then
Zb
Zb
f (x)dx
f (x)dx = lim+
t→a
t
a
if this limit exists (as a finite number)
The improper integrals in (a) and (b) are called convergent if the limit exist and divergent
if the limit does not exist.
Rc
Rb
(c) If f has discontinuity at c (a < c < b), and both f (x)dx and f (x)dx are convergent,
a
then
Zb
Zc
f (x)dx =
a
c
Zb
f (x)dx +
a
f (x)dx
c
Example 3. For what values of p is the integral
4
R1 1
dx convergent?
p
0 x
Example 4. Determine whether each integral is convergent or divergent. Evaluate those
that are convergent.
Z0
1.
√
dx
x+3
−3
Z3
2.
1
√ dx
x x
0
3.
π/2
R
sec2 xdx
π/4
4.
R1
ln xdx
0
5
Comparison theorem Suppose that f and g are continuous functions with f (x) ≥ g(x) ≥
0 for x ≥ a.
R∞
R∞
(a) If f (x)dx is convergent, then g(x)dx is convergent.
a
a
(b) If
R∞
g(x)dx is divergent, then
a
R∞
f (x)dx is divergent.
a
Z∞
Example 5. Use the Comparison Theorem to determine whether
1
or divergent.
6
sin2 x
dx is convergent
x2