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