I am reading about Dirichlet polynomials in the book Analytic Number Theory by the said authors.
During the proof of Theorem 9.1 for any positive real numbers $T, N$ they define a piecewise linear and continuous function $f(t)$ which is an upper bound for the indicator function of     $[0,T]$ and vanishes whenever $t>T+N$ or $t<-N$. Then they use the following bound without comment,
$$
x > 1 
\Rightarrow
\left|
\int_{t \in \mathbb R}f(t) x^{it }\mathrm d t \right|
=O\left( \frac{1}{N (\log x )^2} \right)
.$$

Can anyone justify this inequality?