improper integrals
DESCRIPTION
IMPROPER INTEGRALS. TYPE-II: Discontinuous Integrand Integrands with Vertical Asymptotes. TYPE-I: Infinite Limits of Integration. Examples. IMPROPER INTEGRALS. TYPE-I: Infinite Limits of Integration. TYPE-II: Discontinuous Integrand Integrands with Vertical Asymptotes. - PowerPoint PPT PresentationTRANSCRIPT
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