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:
- Type 1: Infinite Intervals
- 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.
- This works in the opposite as well (from negative infinity to b, provided no discontinuity for t <= b )
- 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.
- Type 2: Discontinuous Integrands
- 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)
- This also works in the opposite way.
- 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