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 `infinity``a``t` 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`