Example IMPROPER INTEGRALS Improper Integral TYPE-I: Infinite Limits of Integration TYPE-I: Infinite Limits of Integration Example TYPE-II: Discontinuous.

Example

IMPROPER INTEGRALS

Improper Integral

TYPE-I:Infinite Limits of Integration

1 2 1 dxx

Example

1

1 2 1 dxx

TYPE-II:Discontinuous IntegrandIntegrands with Vertical

Asymptotes

IMPROPER INTEGRALS

DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1

b

aba dxxf dxxf )(lim)(

Example

1 2 1 dxx

IMPROPER INTEGRALS

DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1

b

aa

b dxxf dxxf )(lim)(

Example

0 dxxex

IMPROPER INTEGRALS

DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1

b

aa

b dxxf dxxf )(lim)(

b

aba dxxf dxxf )(lim)(

The improper integrals

are called convergent if the corresponding limit exists and divergent if the limit does not exist.

a dxxf )(

a dxxf )(

IMPROPER INTEGRALS

DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1

t

ata dxxf dxxf )(lim)(

Example

1 1 dx

x

IMPROPER INTEGRALS

082

IMPROPER INTEGRALS

DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1

dxxf )(

If both improper integrals

are convergent

a dxxf )(

a dxxf )(

convergent

dxxf )(

a

a dxxf dxxf )()(

Example

dx

x 11

2

Memorize:

IMPROPER INTEGRALS

Example

IMPROPER INTEGRALS

Improper Integral

TYPE-I:Infinite Limits of Integration

1 2 1 dxx

Example

1

1 2 1 dxx

TYPE-II:Discontinuous IntegrandIntegrands with Vertical

Asymptotes

IMPROPER INTEGRALS

DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 2

Example

5

2

21 dx

x

IMPROPER INTEGRALS

DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 2

Example

2/

0 sec

dxx

Example

1

0

11 dx

x

IMPROPER INTEGRALS

DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 2

Example

3

0 11 dxx

IMPROPER INTEGRALS

F092

IMPROPER INTEGRALS

Example

1

0 ln dxx

IMPROPER INTEGRALS

IMPROPER INTEGRALS

IMPROPER INTEGRALS

F112

IMPROPER INTEGRALS

IMPROPER INTEGRALS

F092

IMPROPER INTEGRALS

F102