improper integrals

IMPROPER INTEGRALS

TYPE-I:Infinite Limits of Integration

TYPE-I:Infinite Limits of Integration

TYPE-II:Discontinuous Integrand

Integrands with Vertical Asymptotes

TYPE-II:Discontinuous Integrand

Integrands with Vertical Asymptotes

1 2

1 dx

x

0 dxxex

t

t dx

x1 2

1lim

0lim

t

x

t dxxe

dx

x21

1

0

0

Exa

mp

les

IMPROPER INTEGRALS

TYPE-I:Infinite Limits of Integration

TYPE-I:Infinite Limits of Integration

TYPE-II:Discontinuous Integrand

Integrands with Vertical Asymptotes

TYPE-II:Discontinuous Integrand

Integrands with Vertical Asymptotes

3

1 1

1 dx

x

3

1 1

1lim

tt dx

x

1

0 3/2)1(

1 dx

x

t

t dx

x0 3/21 )1(

1lim

2

0 3/2)1(

1 dx

x 2

1

1

0

IMPROPER INTEGRALS

TYPE-I:Infinite Limits of Integration

TYPE-I:Infinite Limits of Integration

TYPE-II:Discontinuous Integrand

Integrands with Vertical Asymptotes

TYPE-II:Discontinuous Integrand

Integrands with Vertical Asymptotes

3

1 1

1 dx

x

3

1 1

1lim

tt dx

x

1

0 3/2)1(

1 dx

x

t

t dx

x0 3/21 )1(

1lim

2

0 3/2)1(

1 dx

x 2

1

1

0

Examples

Memorize:

IMPROPER INTEGRALS

(We need it for chapter10)

(If bigger convg smaller convg)

(If smaller divg bigger divg)

Examples

Compare with 1/x

Examples

IMPROPER INTEGRALS

(If bigger convg smaller convg)

(If smaller divg bigger divg)

Examples

Examples

Remark:

1ln xxex

12 xxx 1for x

ex for Examples

1

2

dxe x

xx ee 2

IMPROPER INTEGRALS

(If bigger convg smaller convg)

(If smaller divg bigger divg)

Examples

Remark:

a

Examples

6

Comparison and limit comparsion not only for Type-I [a,inf) also for Type-II

IMPROPER INTEGRALS

082

IMPROPER INTEGRALS

F092

IMPROPER INTEGRALS

IMPROPER INTEGRALS

F092

IMPROPER INTEGRALS

F102

IMPROPER INTEGRALS

F112