Improper Integral

This extends the concept of a definite integral such that for a function f contains an infinite discontinuity in [a,b], OR where the interval is infinite (ie integrate f in [0,infinity].

Some of these integrations can be done, here are some of the types:

1. Type 1: Infinite Intervals
   1. If the integration of f(x)dx from a to t exists for every number t >= a, then the integration from of f(x)dx from a to infinity = limit as t approaches infinityat of f(x)dx. This requires that the limit exists.
   2. This works in the opposite as well (from negative infinity to b, provided no discontinuity for t <= b )
   3. If both of the above exist, then the integral from negative infinity to infinity is defined as the sum from the integral from negative infinity to zero and zero to infinity.
2. Type 2: Discontinuous Integrands
   1. If f(x) is continuous on [a,b), and is discontinuous at b, then the integral of f(x)dx from on [a,b] = the limit as t approaches b from the negative direction of the integral from a to t of f(x)dx, iff the limit exists (as a finite number)
   2. This also works in the opposite way.
   3. If f(x) has a discontinuity at c, where a < c < b, and both the integration from a to c, and c to b exist, then the integration from a to b = integration from a to c plus integration from c to b