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?

I am reading about Dirichlet polynomials in the book Analytic Number Theory by Iwaniec-Kowalski. 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?