Improper Integrals

1
Improper Integrals

• 8.3

2
Improper Integrals

• There are two types
• One of the limits of the integral is infinite or
  not bounded
• An integral with limits a and b, where the
  function has an infinite discontinuity at c where
  acb

3
Improper Integrals

• Barrons shows samples of integrals that are of
  type 1 and type 2.
• Some integrals fall under both types.

4
Infinite Limit Integrals

• Y e(-x/2)
• Consider the graph of the function above. Now
  consider the area under that curve. You might
  think that the area is infinite, but you will see
  that it is finite.
• First find the integral or area under the curve
  from 0 to some number b. Then we find the limit
  as b approaches infinity of that area formula.
  You will find that this is 2.

5
Improper Integrals with Infinite Integration
Limits

• These are on page 434 in the book.
• If the limit does not exist, then the improper
  integral diverges.
• If the limit exists, then the improper integral
  converges.
• In case 3, both limits on the right need to
  converge for the improper integral to converge.

6
Integrands with Infinite Discontinuities

• Y 1/(x)(1/2) integrated from 0 to 1.
• Since 0 is a point of discontinuity.
• First find the integral or area under the curve
  from some number a to 1. Then we find the limit
  as a approaches 0 from the right of that area
  formula. You will find that this is 2.

7
Integrands with Infinite Discontinuity

• These are on page 436 in the book.
• If the limit does not exist, then the improper
  integral diverges.
• If the limit exists, then the improper integral
  converges.
• In case 3, both limits on the right need to
  converge for the improper integral to converge.

8
Examples

• 4,6,8

9
The p-integrals
http//www.geocities.com/pkving4math2tor6/6_the_in
tgrl/6_07_02_tests_for_conv_of_improper_intgrl.htm

10
Integration

• Basic Integration
• U-substitution
• Do you see a function and its derivative or a
  constant times its derivative.
• Integration by Partsanti-product, usually
  involves ln or trig or inverse trig or exp
• Solve by formula and possibly tabular integration
• What is u? LIPET
• Partial Fraction Integration
• Integral Tables