The document provides a comprehensive overview of improper integrals, detailing their definitions, types, and criteria for convergence and divergence. It discusses various applications of improper integrals across fields such as probability theory, physics, engineering, mathematics, and economics, emphasizing their importance in calculating probabilities, handling infinite distributions, and solving differential equations. Overall, it highlights the critical role of improper integrals in dealing with unbounded intervals and singularities.

1. Improper Integrals

2. Definition of an Improper Integral of Type 1
a) If exists for every number t ≥ a, then
provided this limit exists (as a finite number).
b) If exists for every number t ≤ b, then
provided this limit exists (as a finite number).
The improper integrals and are called
convergent if the corresponding limit exists and divergent
if the limit does not exist.
c) If both and are convergent, then we define

t
a
dx
x
f )
(

b
t
dx
x
f )
(






t
a
a
t
dx
x
f
dx
x
f )
(
)
( lim






b
t
b
t
dx
x
f
dx
x
f )
(
)
( lim


a
dx
x
f )
(


a
dx
x
f )
(


a
dx
x
f )
( 

b
dx
x
f )
(











a
a
dx
x
f
dx
x
f
dx
x
f )
(
)
(
)
(

3. Examples
1
1
1
1
1
1
1
.
1 lim
lim
lim 1
1 2
1 2



























 t
x
dx
x
dx
x t
t
t
t
t
    1
.
2 0
0
0
0
lim
lim
lim 











 

t
t
t
x
t
t
x
t
x
e
e
e
dx
e
dx
e
   
    








































  

2
2
tan
tan
tan
tan
1
1
1
1
1
1
.
3
1
1
0
1
0
1
0
0 2
2
2
lim
lim
lim
lim
t
t
x
x
dx
x
dx
x
dx
x
t
t
t
t
t
t
All three integrals are convergent.

4.     













 1
ln
ln
ln
1
1
lim
lim
lim 1
1
1
t
x
dx
x
dx
x t
t
t
t
t
An example of a divergent integral:
The general rule is the following:
1
p
if
divergent
and
1
p
if
convergent
is
1
1




dx
xp


1 2
convergent
is
1
that
slide
previous
the
from
Recall dx
x

5. Definition of an Improper Integral of Type 2
a) If f is continuous on [a, b) and is discontinuous at b, then
if this limit exists (as a finite number).
a) If f is continuous on (a, b] and is discontinuous at a, then
if this limit exists (as a finite number).
The improper integral is called convergent if the
corresponding limit exists and divergent if the limit does
not exist.
c) If f has a discontinuity at c, where a < c < b, and both
and are convergent, then we define

 


t
a
b
a
b
t
dx
x
f
dx
x
f )
(
)
( lim

 


b
t
b
a
a
t
dx
x
f
dx
x
f )
(
)
( lim

b
c
dx
x
f )
(

c
a
dx
x
f )
(

b
a
dx
x
f )
(


 

b
c
c
a
b
a
dx
x
f
dx
x
f
dx
x
f )
(
)
(
)
(

6. Example 1:
1
0
dx
x

1
0
lim ln b
b
x


0
lim ln1 ln
b
b


  This integral is divergent.
1
0
1
lim
b
b
dx
x

 
    1
1
2
1
2
1
lim
lim
lim 0
1
0
1
0
1
0












 t
x
dx
x
dx
x t
t
t
t
t
Example 2:
This integral is convergent.

7. APPLICATION OF IMPROPER INTEGRALS
1.Probability Theory
.Probability Distribution: improper integrals
used to calculate probabilities for continuous
random variables especially for distribution
defined over infinite intervals such as normal
and exponential distribution.
.Expected Values :improper integrals are used to
calculate values, and probabilities distributions.

8. PHYSICS
.Electrostatics and gravitational
These are created by continuous charge or mass
distributions are often calculated using improper
integrals especially when the distribution extends over
an infinite range.
.Quantum mechanics: Improper integrals arise
in at wavefunctions which often extended to infinity.
Heat transfer and diffusion : The solution to the heat
equation or diffusion often involves improper integrals.

9. Engineering
.Signal processing : Fourier transforms and
transforms both essential tools in signal
processing involve improper integrals over
infinite intervals .These transforms are used to
analyze signals and systems in the frequency
domain.
.Control systems : improper integrals are used
to analysis of control systems .

10. Mathematics
• Area and volume calculations : improper integrals are
used to compute areas under curves and volumes of
solids of revolution especially when the region extends
to infinity.
• .Special functions : many special functions like the
gamma function and beta function are defined using
improper integrals.
• Convergence and divergence : improper integrals are
used to study the behavior of functions particularly in
the analysis of series and limits.

11. Economics
• Discounted cash flow : In continuous time finance
the method uses improper integrals to calculate the
present value of future cash flows when the time
horizon is infinite.
• Conclusion:
• Improper integrals are crucial in fields requiring
integration over unbounded intervals or where the
function has singularities making them indispensable
in both theoretical and applied contexts.

12. THANK YOU
• 24H51A0520
• D.HAREESH