There's a problem (#10.23) in Apostol's Mathematical Analysis of which I am having a rough time solving: Let $F(y)= \int_{0}^{\infty}\frac{\sin xy}{x(x^{2}+1)}dx$ if $y > 0$. Show that $F$ satisfies the differential equation $F''(y)-F(y)+\frac{\pi }{2} = 0$ and deduce that $F(y)= \frac{1}{2}\pi(1-e^{-y})$. Use this result to deduce the following equations, valid for $y > 0$ and $a > 0$: Use this to deduce that for $y>0$ and $a>0$ \begin{align} \int\nolimits_{0}^{\infty}\frac{\sin xy}{x(x^{2}+a^{2})}dx = \frac{\pi}{2a^{2}}(1-e^{-ay}), \newline \int_{0}^{\infty}\frac{\cos xy}{x^{2}+a^{2}}dx = \frac{\pi e^{-ay}}{2a}, \newline \int_{0}^{\infty}\frac{x\sin xy}{x^{2}+a^{2}}dx = \frac{\pi}{2}e^{-ay} \end{align} We could use $\int_{0}^{\infty}\frac{\sin x}{x}dx = \frac{\pi}{2}$

How can I show this?