Hello,

For one of the integration questions I have to do, I need to show that

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

is Lebesgue integrable, where1is the indicator function, and that

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

Does anyone have any ideas about how I would solve this? Thanks for any help!