When I first learnt the power rule for integration in school, namely
it always bugged me how the special case of required separate treatment. In my mind, there was no reason for
to not work.
Many years later, it suddenly occurred to me that it does indeed work; only the constant of integration must be infinite. Specifically, plus something finite:
The bracketed portion is assuredly , as it vanishes when . As a side-effect, we immediately see why a logarithm grows slower than any polynomial: because a logarithm is a polynomial with infinitesimal degree.
If all of this is sacrilege unto you, consider reading some Euler (see translations by Ian Bruce at <www.17centurymaths.com>).
If you're still not convinced, compute
and be satisfied when you get at the end.
Conway (2022). The power rule for integration and the logarithm. <https://yawnoc.github.io/math/power-rule-log> Accessed 2025-06-15.