How is this derived? In my textbook I find the following derivation:
$$ \displaystyle \lim _{n \to \infty} \dfrac{1}{n} \displaystyle \sum ^n _{k=1} \dfrac{1}{1 + k/n} = \displaystyle \int^1_0 \dfrac{dx}{1+x}$$
I understand that it's $\displaystyle \int^1_0$ but I don't understand the $\dfrac{dx}{1+x}$ part. 

The sum is a Riemann Sum for the given integral. As $n\to\infty$,
$$
\sum_{k=1}^n\frac1{1+k/n}\frac1n
$$
tends to the sum of rectangles $\frac1{1+k/n}$ high and $\frac1n$ wide. This approximates the integral
$$
\int_0^1\frac1{1+x}\mathrm{d}x
$$
where $x$ is represented by $k/n$ and $\mathrm{d}x$ by $\frac1n$.
