I have noted the value of one in the derivatives and wondered if it's fundamental then in my study I encountered several examples of this sort:
$$ \int \frac {dx}{a^{2} + x^{2} } = \frac {1}{a} \tan ^{-1}{\left(\frac{x}{a}\right)}, $$
which I had no problem proving.
The intuition I developed was that we could derive a relation between the integrals of the divisions of an angle and the number of divisions.
Questions that arose:
- Is it not possible to simply divide the angle and get the integral for the result like it's an angle on its own or is there a certain significance?
- Does $a $ have to be an integer? Could it be a function or is that nonsense (and why is it nonsense)?
This is just a change of variable. Consider $$I=\int \frac {dx}{a^{2} + x^{2} } $$ Let $x=a y$, $dx=a \,dy$ to make $$I=\int \frac {a\,dy}{a^{2} +a^2 y^{2} }=\frac 1a \int \frac {dy}{1 + y^{2} }$$ $a$ being anything you want (integer, rational, irrational, complex, ...) as long as it is not $0$.