Could you please help me with this:

show that for -T<t<T, the function $\displaystyle f_{t} : \mathbb{R} \rightarrow \mathbb{R} $ defined by

$\displaystyle f_{t} (x) = \frac{e^{-x} (1-cos(tx))}{x^2} \bold{1}_{ [0, \infty) } (x) $

is Lebesgue integrable and that

$\displaystyle \int_{0}^{\infty} \frac{e^{-x} (1-cos(tx))}{x^2} dx = t .tan^{-1} t - \frac{1}{2} log (1+t^{2}) $

We are allowed to use results from Lebesgue integration.

Thanks in advance!