Why didn’t we use the integral approximation (A.12) directly on $\sum_{k = 1}^n 1/k$ to obtain an upper bound on the $n$-th harmonic number?.

To get an upper bound using integral approximation (A.12), we need to integrate the function from $x = (1 - 1) = 0$. This makes the function $\frac 1 x$ undefined because of division-by-zero. To avoid this, we took out the first term ($k = 1$) and carried out the sum from $k = 2$ and the integral from $x = 1$.