The Logarithmic Black Sheep
Behold, the integral power rule: For any real $p$, $$\int x^p\, dx = \frac{1}{p+1} x^{p+1} +C$$ Dependable. Ubiquitous. Cursed with an asterisk: ... unless $p=-1$, in which case $$\int x^p\, dx = \ln(x)+C.$$ I have gotten used to this fact through years of working with integrals. All the same, it bothers me. Every time I am forced to check if my exponent falls into this one exceptional case, a voice somewhere in my head protests. "How does this make sense?? How do you get a continuous family of monomials except for the one case where, of all things, you get a logarithm!?" This is not to say I don't understand why $\int dx/x$ spits out a logarithm. I can see that the usual formula would force you to divide by 0 at $p=-1$, and I could even prove the antiderivative in a few lines. It's just that it feels like an unexplained discontinuity at a point. I never learned how this logarithmic black sheep fits in with the big happy family of monomials. Fitting the