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.
1. Definition of an improper integral of type 1 (infinite intervals)
(a) If
∫t
f (x)dx exists for every number t ≥ a, then
a
∫t
∫∞
f (x)dx
f (x)dx = lim
t→∞
a
a
provided this limit exists (as a finite number)
(b) If
∫b
f (x)dx exists for every number t ≤ b, then
t
∫b
∫b
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.
∫∞
∫b
(c) If both f (x)dx and
f (x)dx are convergent, then we define
a
−∞
∫∞
∫a
f (x)dx =
−∞
∫∞
f (x)dx +
−∞
f (x)dx
a
where a is any real number
Example 1. For what values of p is the integral
1
∫∞ 1
dx convergent?
p
1 x
Example 2.
Determine whether each integral is convergent or divergent. Evaluate
those that are convergent.
∫∞
(a)
dx
√
x+3
2
∫3
(b)
−∞
(c)
∫∞
−∞
∫∞
(d)
dx
+9
x2
(2x2 + x − 1)dx
1
dx
(x + 2)(x + 3)
0
2
(e)
∫∞
ex dx
1
(f)
∫1
ex dx
−∞
(g)
∫∞
ex dx
−∞
∫∞
(h)
dx
x(ln x)2
e
3
2. Definition of an improper integral of type 2 (discontinuous integrands)
(a) If f is continuous on [a, b) and is discontinuous at b, then
∫b
∫t
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
∫b
∫b
f (x)dx = lim+
f (x)dx
t→a
a
t
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.
∫c
∫b
(c) If f has discontinuity at c (a < c < b), and both f (x)dx and f (x)dx are convera
gent, then
∫c
∫b
f (x)dx =
Example 3. For what values of p is the integral
4
∫b
f (x)dx +
a
a
c
f (x)dx
c
∫1 1
dx convergent?
p
0 x
Example 4. Determine whether each integral is convergent or divergent. Evaluate those that
are convergent.
∫0
1.
−3
∫3
2.
dx
√
x+3
1
√ dx
x x
0
3.
π/2
∫
sec2 xdx
π/4
4.
∫1
ln xdx
0
5
Comparison theorem. Suppose that f and g are continuous functions with f (x) ≥ g(x) ≥ 0
for x ≥ a.
∫∞
∫∞
(a) If f (x)dx is convergent, then g(x)dx is convergent.
a
(b) If
∫∞
a
g(x)dx is divergent, then
a
∫∞
f (x)dx is divergent.
a
∫∞
Example 5.
Use the Comparison Theorem to determine whether
1
or divergent.
6
sin2 x
dx is convergent
x2