Definition of $e$
$\ln (e)= \int_\limits{1}^e\frac{1}{t}dt=1$
I found this in my calculus textbook, and I was wondering what does this exactly "mean". This is more of a research question which I could do, but where would I start. I understand that its an integral, and what it states; however, how was this derived in calculus? Where would be a good book to research to understand an ideal proof of the equation?
To say that $e$ is the number such that $\int_\limits{1}^e\frac1x\mathrm{d}x=1$ is to say that when you graph the curve of $y=\frac1x$ and shade in the area to the right of $x=1$, then $e$ is the boundary on the right such that the shaded area is exactly $1$. Take a look at the diagram below. The green curve is $y=\frac1x$ and $a$ is the area of the brown region.
If the boundary on the right were $x=3$, we'd have slightly too much area. If it were instead $2$, we'd have too little area. But $e=2.718\ldots$ is the point that's just right.