
Integral
calculus 


The improper integrals

The improper integral definition

The improper integrals examples






The
improper integrals

A
definite integral with one or both limits of integration infinite,
or having an integrand that becomes infinite between
the limits of integration is called the improper
integral.

Let
f (x)
be defined on [a,
oo
)
and integrable on [a,
b] for all a
< b <
oo,
then


provided
the limit exists.

If
f (x)
is defined on ( 
oo,
b] and integrable on [a,
b] for all 
oo
< a < b,
then


provided
the limit exists.

If
f (x)
is defined on ( 
oo,
oo
)
and integrable on any closed interval [a,
b], then


provided
the limits on the right exist.

In
each case where the appropriate limit exists we say the integral
converges, otherwise the integral is said to diverge

.


Example:
Evaluate 




As
the area of the region beneath the graph of f
over the interval [1,
b] as b
goes to infinity A
= 1, thus the
integral
converges.

Example:
Evaluate 



Therefore,
the integral diverges.


Example:
Evaluate 




Therefore,
the integral converges.


Example:
Evaluate 




Therefore,
the integral converges.


Example:
Evaluate 




Therefore,
the integral converges.


Example:
Evaluate 




Therefore,
the integral converges.









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