Last week I discovered a very neat property of improper Integrals. There are some functions where it is possible to calculate the area under the curve with the Fundamental Therorem of Analysis even if the function is discontinuous .
Let's say you have a function and you want to calculate . But unfortunately there is a pole point at with .Now I will prove that: If is the only point on where is discontinuos and there exists a point to which is point symmetric, then it is possible to calculate the integral with the Fundamental Therorem of Analysis.
Because it is an improper integral, you devide the ingral into two:
Because is the only point of discontinuity of , is continous on and on and therefore you can calculate this with the Fundamental Therorem of Analysis, if is the antiderivative of :
If is a pole point of then it is a pole point of as well. is point symmetric to and because is a measure of the slope of (because ), must be axially symmetrical to .
an example point symmetric function in green and a possible axial symmetrical antiderivative of in blue
The geometric interpretation of this relationship is, that the infinite areas on the left and right side of the pole cancel out each other because of the opposite sign.
It is important to mention, that the antiderivative must be defined for every in , because otherwise it will not work.