Need help understanding where this formula is derived from I was reading the book interesting integrals and this came up:
$$\int_{j}^{j+1} \frac{n-j}{x} \mathrm{d}x$$ it then goes on to say that $j$ is equal to floor $x$ because the integration interval $j \leq x < j+1$. I get this but aren't the limits of integration on the interval $j\leq x\leq j+1$?

It's irrelevant. Adding or removing a single point will not change the integral; if you think about it, a single point will belong to a single very small division in the partition of the interval, so its contribution to the value of the integral is zero.  
