Taylor series leads to two different functions - why?

taylor-expansion calculus

When you integrate, you should include a constant of integration. What you see here is that when integrating the functions, you get different constants of integration. This is why your answers differ by only a constant, namely $a\ln a$ (you can see this by use of $\log$ rules).

If you take care with the limits or boundary conditions in the integration step, then the answers will agree exactly.

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