7.8 improper integrals definition: the definite integral is called an improper integral if (a) at...
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7.8 Improper Integrals
Definition:
The definite integral is called an improper integral if
(a) At least one of the limits of integration is infinite, or
(b) The integrand f(x) has one or more points of discontinuity on the interval [a, b].
b
adxxf )(
Infinite Limits of Integration
1. If f is continuous on the interval , then
2. If f is continuous on the interval , then
3. If f is continuous on the interval , then
),[ a
In each case, if the limit exists, then the improper integral is said to converge; otherwise, the improper integral diverges. In the third case, the improper integral on the left diverges if either of the improper integrals on the right diverges.
),(
],( b
a
b
abdxxfdxxf )(lim)(
b b
aadxxfdxxf )(lim)(
c
cdxxfdxxfdxxf )()()( where c is any
real number
Discontinuous Integrand1. If f is continuous on the interval and has an
infinite discontinuity at b, then
2. If f is continuous on the interval and has an infinite discontinuity at a, then
3. If f is continuous on the interval except for some c in (a, b) at which f has an infinite discontinuity, then
),[ ba
In each case, if the limit exists, then the improper integral is said to converge; otherwise, the improper integral diverges. In the third case, the improper integral on the left diverges if either of the improper integrals on the right diverges.
],[ ba
],( ba
b
a
c
abcdxxfdxxf )(lim)(
b
a
b
cacdxxfdxxf )(lim)(
b
c
b
a
c
adxxfdxxfdxxf )()()(
Examples:
Determine whether each integral is convergent or divergent. Evaluate those that are convergent.
1)
323
2lim3
23
2lim
32lim32lim
323232
0
23023
0
20 2
0
0
222
b
ba
a
b
baa
xxx
xxx
dxxxdxxx
dxxxdxxxdxxx
Now,
3
23
2300lim3
23
2lim
23
023 aa
xxa
aa
diverges 32 2
dxxx
2) dxexdxxe xt
t
x
00
lim
x
x
x
e
e
ex
0
1
D I
11
lim
0lim
lim
0
0 0
tt
t
tt
t
txx
t
x
ete
eete
exedxxe
Note: Rule) sHospital'L' UseForm, nate(Indetermi lim
t
tte
01
limlimlim
tttt
t
t ee
tte
Thus, 110011
lim
tt
t ete
1. toconverges 0 dxxe x
3)
diverges )(sec
1)tan(lim)tan(lim
)(seclim)(sec
2/
4/
2
2/4/
2/
4/
2
2/
2/
4/
2
dxx
bx
dxxdxx
b
b
b
b
b
4)
t
t
s
sdx
xxdx
xx
dxxx
dxxx
0202
4
0
4
0 2
)3)(2(
1lim
)3)(2(
1lim
)3)(2(
1
6
1
Now,
5/1 ,5/1
)2()3(1
32)3)(2(
1
BA
xBxA
x
B
x
A
xx
4
20
2
4
202
3ln5
12ln
5
1lim3ln
5
12ln
5
1lim
)3(51
)2(51
lim)3(
51
)2(51
lim
tt
s
s
tt
s
s
xxxx
dxxx
dxxx
where
)3ln(5
1)2ln(
5
13ln
5
12ln
5
1lim3ln
5
12ln
5
1lim
20
2sxxx
s
s
s
4
0 2diverges
6
1dx
xx
Comparison Theorem
Suppose that f and g are continuous functions with
for
(a) If is convergent, then
is convergent,
(b) If is divergent, then
is divergent
0)()( xgxf ax
adxxf )(
adxxg )(
adxxg )(
adxxf )(
We use the Comparison Theorem at times when its impossible to find the exact value of an improper integral.
Examples:Use the Comparison Theorem to determine whether the integral is convergent or divergent.
1) dxx
x
1
1
Observe that
),1[on 11
xx
x
Now,
divergent. is 1
12
111
1
1 2/11
dxx
pdxx
dxx
Note:
Theorem. Comparison by thedivergent is 1
1dx
x
x
Thus,
1 ifdivergent and
1 if convergent is 1
1
p
px p
2) 1
0dxx
e x
Observe that
10for 1
1
xxx
e
ex
x
Now,
222lim
2lim1
lim1
0
11
0
2/1
0
1
0
a
xdxx
dxx
a
aaaa
Theorem. Comparison by the convergent is 1
0
dxx
e x