Integral calculus
The improper integrals
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
 Solution:
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
 Solution:
Therefore, the integral diverges.
 Example:   Evaluate
 Solution:
Therefore, the integral converges.
 Example:   Evaluate
 Solution:
Therefore, the integral converges.
 Example:   Evaluate
 Solution:
Therefore, the integral converges.
 Example:   Evaluate
 Solution:
Therefore, the integral converges.
Functions contents G