I just read a proof which argued that since the derivative of $\ln(x)$ is $1/x$, it follows from the Fundamental Theorem of Calculus that $\ln(x) = \int_1^x 1/t dt$.
Now, supposing it is indeed the case that
$$ \frac{d}{dx} \ln(x) = \frac{1}{x} $$
How do you know, using the fundamental theorem of calculus, that
$$ \ln(x) = \int_1^x \frac{1}{t} d? $$
That is, how do you know that the bounds of the integral should be from $1$ to $x$ as opposed to any other $a \in \mathbb{R}$ to $x$?