c Dr Oksana Shatalov, Fall 2012
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8.9: Improper Integrals
TYPE I: Infinite Interval and Continuous Integrand
Z ∞
1
EXAMPLE 1. Evaluate
dx
x2
1
• What is the area, A, under the curve y =
• What is the area, At , under the curve y =
1
x2
1
x2
on [1, ∞) is?
on [1, t), t > 1, is?
y
x
0
REMARK 2. Not all areas on an unbounded interval will yield finite areas.
DEFINITION 3. An improper integral is called convergent if the associated limit exists and is a finite
number. An improper integral is called divergent if the associated limit does not exist or is −∞, or ∞.
Z ∞
1
EXAMPLE 4. Evaluate
dx
x
1
Z
FACT: If a > 0 then
a
∞
1
dx is convergent as p > 1 and divergent as p ≤ 1.
xp
c Dr Oksana Shatalov, Fall 2012
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How to deal with Type I Improper Integrals:
Z t
• If
f (x) dx exists for every t ≥ a then
a
∞
Z
Z
t
f (x) dx = lim
t→∞ a
a
f (x) dx
provided the limit exists and finite.
Z b
• If
f (x) dx exists for every t ≤ b then
t
Z
b
Z
f (x) dx = lim
t→−∞ t
−∞
b
f (x) dx
provided the limit exists and finite.
Z c
Z ∞
• If
f (x) dx and
f (x) dx are BOTH convergent then
−∞
c
Z
∞
Z
c
f (x) dx =
−∞
where c is any number.
Z
0
EXAMPLE 5. Evaluate I =
−∞
√
1
dx
20 − x
Z
f (x) dx +
−∞
∞
f (x) dx
c
c Dr Oksana Shatalov, Fall 2012
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Z
∞
EXAMPLE 6. Evaluate I =
−∞
2
xe−x dx
c Dr Oksana Shatalov, Fall 2012
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Z
∞
sin x dx
EXAMPLE 7. Evaluate I =
−2
TYPE II: Discontinuous Integrand and Finite Interval
• If f (x) is continuous on [a, b) and not continuous at x = b then
b
Z
t
Z
f (x) dx = lim
t→b−
a
f (x) dx
a
if the limit exists and finite.
• If f (x) is continuous on (a, b] and not continuous at x = a then
Z
b
Z
f (x) dx = lim
t→a+
a
b
f (x) dx
t
if the limit exists and finite.
Z
• If f (x) is continuous on [a, c) and (c, b] and not continuous at x = c, and the integrals
Z b
and
f (x) dx are both convergent then
c
Z
b
Z
f (x) dx =
a
c
Z
f (x) dx +
a
b
f (x) dx.
c
c
f (x) dx
a
c Dr Oksana Shatalov, Fall 2012
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Z
10
√
EXAMPLE 8. Evaluate I =
0
Z
1
dx
10 − x
1
ln x dx
EXAMPLE 9. Evaluate I =
0
c Dr Oksana Shatalov, Fall 2012
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Z
3
EXAMPLE 10. Evaluate I =
−2
1
dx
x3
Now we consider an integral involving both of these cases.
Z ∞
1
EXAMPLE 11. Evaluate I =
dx
x2
0
Comparison Theorem: Suppose f (x) and g(x) are continuous functions s.t f (x) ≥ g(x) ≥ 0 for
x ≥ a. Then
Z ∞
Z ∞
1. if
f (x) dx is convergent then
g(x) dx is convergent;
a
Z
2. if
a
∞
Z
g(x) dx is divergent then
a
∞
f (x) dx is divergent.
a
c Dr Oksana Shatalov, Fall 2012
EXAMPLE 12. Determine whether the following integrals are convergent or divergent.
∞
Z
(a) I =
1
∞
Z
(b) I =
1
Z
(c) I =
1
∞
sin2 x
dx
x2
p
√
1+ x
√
dx
x
1
dx
x + e2x
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