Section 7.7 Improper Integrals

• So far in computing definite integrals on the interval a ≤ x ≤ b, we have assumed our interval was of finite length and our function was continuous

• This is not always the case

• We have to be able to compute integrals that are on unbounded intervals

• We need to be able to compute integrals of functions that may not be continuous on our given interval

• Such integrals are called improper

I. When the limit of integration is infinite

• Consider

• We calculate

• Now we take the limit as b ∞

• So we say converges to 1

1 2

1dx

x

1

1111

11 2

bdx

xx

bb

111

lim1

lim1 2

bdx

x b

b

b

1 2

1dx

x

• Suppose f(x) is positive for x ≥ a.

If is a definite number, we say that

converges.

Otherwise it diverges.

b

abdxxf )(lim

adxxf )(

Examples

0 2 1

2dx

x

x

1

1dx

x

0dxxex

Recall

• If either of the integrals diverges, the whole thing diverges

c

cdxxfdxxfdxxf )()()(

II. When the integrand becomes infinite

• In this case we may have a finite interval, but the function may be unbounded somewhere on the interval

• Consider

• Has an asymptote at x = 0

• Handle it in a similar way

1

0

1dx

x

• We compute

• Now we take the limit

• So converges to 2

2/1122

1adx

xa

222lim1

lim 2/1

0

1

0

adxx aaa

1

0

1dx

x

• Suppose f(x) is positive for x ≥ a.

If is a definite number, we say that

converges.

Otherwise it diverges.

As before we can split the interval so the point of discontinuity is an end point

c

abcdxxf )(lim

b

adxxf )(

Examples

4

0 2

2

x

dx

2

0ln dxx