Last time we looked at integrals which weren’t proper integrals because the limits of integration were infinite (either on one side, or both). This was one of the constraints we had on a well defined Riemann sum. The other constraint we had was that there were no infinite discontinuities in the integrand. Here we will show that sometimes we can indeed define an improper integral which does include such a discontinuity within the limits of integration.
Improper Integrals of the second kind: Infinite discontinuities in the integrand
We have seen what happens when we integrate from or to . Sometimes it gives a convergent value and we can find the integral, sometimes the improper integral does not converge and it gives us an answer of . Indeed sometimes it doesn’t blow up, but it just doesn’t converge – as in the integral of over an infinite range.
Now we are going to look at what happens if the function itself has an infinite discontinuity in it and we want to integrate over this region, or include it at one of the limits of integration.…