We start with a funciton, integrate it, expand it into a power series, then differentiate to get back to the original function. We can convert the new found series into a polynomial and evaluate it to demonstrate that it take values close to original funciton. We can use the same trick in reverse too. Take a function, differentiate it, expand into a power series, and integrate. When we integrate, the constant of integration will become the constant term of the series.
Lets consider an example.
We start with a funciton, integrate it, expand it into a power series, then differentiate to get back to the original function. Every indefinite integral has a constant of integration. In particular, when we integrate the series, we can't disregard the constant of integration. It turns out to be constant term of the series. Lets look at an example.