TITLE: How to find the indefinite integral of $\frac{1}{x}$?
QUESTION [2 upvotes]: The indefinite integral of $\frac{1}{x}$ is $\ln x$, but I only know this because I know that the derivative of $\ln x$ is equal to $\frac{1}{x}$. 

Is there a way to find that the integral of $\frac{1}{x}$ is $\ln x$, without using the knowledge of the derivative of $\ln x$?

REPLY [0 votes]: It is difficult to avoid circularity when addressing this question.
Anyway, you can remark that by a rescaling of the variable
$$L(x):=\int_1^x\frac{dx}x=\int_y^{xy}\frac{y\,dx}{y\,x}=\int_y^{xy}\frac{dx}x.$$
This can be written
$$L(xy)=L(x)+L(y),$$ as $L(1)$ is obviously $0$.
This gives you a famous functional equation.

From this, by induction
$$L(x^n)=nL(x),$$ then $$L(x^{n/m})=\frac nmL(x),$$ and by extension to the reals
$$L(x^y)=yL(x).$$ 
Now there must be a constant, let $e$, such that $L(e)=1,$
 then
$$L(e^x)=x$$
and  $L$ appears to be the inverse of an exponential.
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