Lemma 4.7: (p74) Let $f(x)$ be a real function with a continuous and steadily increasing derivative $f'(x)$ in $(a,b)$, and let $f'(b) = \alpha$, $f'(a) = \beta.$ Then, $$ \sum_{a < n \le b } e^{2 \pi i f(n) } = \sum_{\alpha - \eta < \nu < \beta + \eta} \int_a^b e^{2 \pi i (f(x) - \nu x ) } \, dx + O\{ \log (\beta - \alpha + 2 ) \}, $$ where $\eta$ is any positive constant less than $1$. (The Theory of the Riemann Zeta-function, Titchmarsh)
The text proves the following two results:
By making change of variables $h(x)=f(x)-kx$ , we may assume wlog $\eta -1 < \alpha \le \eta.$
$$ \sum_{a < n \le b } e^{2 \pi i f(n) } = \sum_{\alpha - \eta < \nu < \beta + \eta} \int_a^b e^{2 \pi i (f(x) - \nu x ) } \, dx +O(\log (\beta+2)) + O(1). $$ "The result therefore follows".
My question is what happened to "$ \beta - \alpha+2$" in the original expression? Why does "the theorem follows" - or is there a typo?