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Math 152, Benjamin
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8.9 Improper Integrals
An integral is said to be “improper” when either:
• The interval over which you are integrating is infinite.
• The function has a discontinuity somewhere in the interval over which you are integrating.
Type 1: Infinite Intervals
On an interval [a, ∞]:
Z
∞
f (x) dx = lim
Z
t
t→∞ a
a
f (x) dx
On an interval [−∞, b]:
Z
b
f (x) dx = lim
Z
t→−∞ t
−∞
b
f (x) dx
The integral is said to converge if the limit exists and diverge if the limit does not exist.
On the interval [−∞, ∞], we split up the integral into two separate improper integrals.
Z
∞
f (x) dx =
−∞
Z
a
f (x) dx +
Z
∞
f (x) dx
a
−∞
The integral converges as long as BOTH integrals on the right converge.
If EITHER integral on the right diverges, then the original integral diverges.
Determine whether the following integrals converge or diverge. If the integral converges, find its value.
•
Z
•
Z
∞
1
dx
x2
∞
1
dx
x
1
1
Z
1
dx converges ONLY when p > 1. If p ≤ 1, it diverges.
xp
1
The lower limit of integration does not have to be 1. This is actually true if the lower limit of integration
is any positive number greater than 0.
Fact of Mucho Importance:
∞
1
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Math 152, Benjamin
Aurispa
•
Z
•
Z
•
Z
∞
1
∞
ln x
dx
x
3
x2 e−x dx
0
0
xex dx
−∞
2
c
Math 152, Benjamin
Aurispa
•
Z
•
Z
∞
1
dx
x2 − 7x + 12
∞
1
dx
x2 + 9
5
−∞
3
c
Math 152, Benjamin
Aurispa
Type 2: Discontinuous Functions on an Interval [a, b]
If f (x) is continuous everywhere on the interval except at x = b, then
Z
b
f (x) dx = lim
Z
t
f (x) dx
t→b− a
a
If f (x) is continuous everywhere on the interval except at x = a, then
Z
b
f (x) dx = lim
t→a+
a
Z
b
f (x) dx
t
The integral is said to converge if the limit exists and diverge if the limit does not exist.
If f (x) has a discontinuity at x = c somewhere in the interval (a, b), then we split up the integral into two
improper integrals.
Z
b
f (x) dx =
Z
c
a
a
f (x) dx +
Z
b
f (x) dx
c
The integral converges as long as BOTH integrals on the right converge. If EITHER integral on the right
diverges, then the original intergral diverges.
•
Z
•
Z
2
0
1
dx
(x − 2)5
5
−3
√
1
dx
x+3
4
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Math 152, Benjamin
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•
Z
•
Z
6
−4
7
dx
(x + 2)2
2
x ln x dx
0
Comparison Test for Integrals
Suppose that f and g are positive continuous functions for all x ≥ a
R
• If f (x) ≤ g(x) and if a∞ g(x) dx converges, then
so does the smaller.
• If f (x) ≥ g(x) and if
so does the larger.
R∞
a
g(x) dx diverges, then
R∞
f (x) dx also converges. If the bigger converges,
R∞
f (x) dx also diverges. If the smaller diverges,
a
5
a
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Math 152, Benjamin
Aurispa
Determine whether the following integrals converge or diverge by comparison with an appropriate integral
(if possible).
•
Z
•
Z
•
Z
•
Z
∞
1
6
dx
x4 + 1
∞
√
2
∞
5 + e−x
dx
x
∞
cos2 x
√ dx
x x
3
2
x2
dx
x5 − 1
6
c
Math 152, Benjamin
Aurispa
•
Z
•
Z
∞
sin x + 4
dx
x
∞
1
dx
x + e2x
2
1
7