Math 1B
§ 7.8 Improper Integrals
Overview: In this section we extend the concept of a definite integral to the case where the interval is
infinite and to the case where f has an infinite discontinuity in [ a,b] .
I. Infinite Intervals
Consider f ( x ) =
1
. Integrate f ( x ) over [1,t ] :
x2
€
€
€
€
t
∞
∫ f ( x )dx = lim ∫ f ( x )dx
Definition:
t →∞
a
a
∞
If the limit exists,
∫ f ( x )dx is convergent.
a
If€the limit does not exist,
∞
∫ f ( x )dx is divergent.
a
€
∞
Example: Evaluate €∫
1
1
dx .
x
€
Stewart – 7e
1
∞
Example: Evaluate
∫ (1− x )e
−x
dx .
0
€
b
∫
Definition:
b
f ( x ) dx = lim
t →−∞
−∞
∫ f ( x )dx
t
b
If the limit exists,
∫ f ( x )dx is convergent.
−∞
If€the limit does not exist,
b
∫ f ( x )dx is divergent.
−∞
€
Stewart – 7e
2
€
2
Example: Evaluate
∫x
−∞
2
1
dx .
+4
€
c
∞
Definition:
∞
∫ f ( x )dx = ∫ f ( x )dx + ∫ f ( x )dx
−∞
where c is any real number (usually 0).
c
−∞
c
∞
∞
∫ f ( x )dx is convergent if both ∫ f ( x )dx and ∫ f ( x )dx converge, and is divergent otherwise.
€
−∞
c
−∞
∞
Example: Evaluate
€
∫
−∞
x
2
x€+ 2
dx
€
€
Stewart – 7e
3
II. Discontinuous Integrands
3
Consider the integral
∫
0
1
( x − 2)
2
dx .
3
This integral is improper because the integrand is undefined at x = 2 (vertical asymptote at x = 2 ).
b
Definition:€ If f is continuous on [ a,b) and is discontinuous at b, then
∫
t
f ( x ) dx = lim
t →b−
a
b
If f is continuous on ( a,b] and is discontinuous at a, then
3
Example: Evaluate
∫
2
a
b
∫ f ( x )dx = lim ∫ f ( x )dx .
t →a +
a
€
∫ f ( x )dx .
t
€
1
dx .
3€
−x
€
€
c
Definition: If f has a discontinuity at c, a < c < b , and both
b
∫ f ( x )dx and ∫ f ( x )dx are convergent,
a
b
then
c
∫ f ( x )dx = ∫ f ( x )dx + ∫ f ( x )dx .
a
a
c
€
Stewart – 7e
€
c
b
€
4
4
Example: Evaluate
∫
1
1
( x − 2)
2
dx .
3
€
2
Note: If we confused the integral
∫
0
1
with an ordinary integral, we get:
2 dx
( x −1)
€
Stewart – 7e
5
Sometimes we can’t find the exact value of an improper integral but we can determine whether it is
convergent or divergent.
Comparison Theorem: Suppose that f and g are continuous functions with g( x ) ≤ f ( x ) 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 ∫ f ( x )dx is divergent.
a
€
Useful Fact:
€
a
∞
∫
1
€
1
dx is convergent if p > 1 and divergent if p ≤ 1. (Example 4, page 522)
xp
€
∞
x2
dx is convergent
Example: Determine whether €∫ 5
€ or divergent.
x
+
1
0
€
€
Stewart – 7e
6