Lebesgue and Riemann integral

Question

I'm asked to discuss the existence of Riemann and Lebesgue integral of the function: $$ F(t)= \int ^{\infty}_0 e^{-xt}\dfrac{\sin x}{x}dx, \quad t\geq 0. $$

Please, I need the steps to follow, a routemap to discuss it

Answer 1

We know that $\left|\dfrac{\sin x}{x}\right|\leq 1$ for $x\ne 0$ so

\begin{align*} \int_{0}^{\infty}\left|e^{-xt}\frac{\sin x}{x}\right|dx\leq\int_{0}^{\infty}e^{-xt}dx=\frac{1}{t}<\infty \end{align*} so the Lebesgue integral exists. In particular, by Lebesgue Dominated Convergence Theorem we have \begin{align*} \lim_{u\uparrow\infty,~v\downarrow 0}\int_{v}^{u}e^{-xt}\frac{\sin x}{x}dx=\lim_{u\uparrow\infty,~v\downarrow 0}\int_{0}^{\infty}\chi_{[v,u]}(x)e^{-xt}\frac{\sin x}{x}dx \end{align*} exists and equals to $\displaystyle\int_{0}^{\infty}e^{-xt}\dfrac{\sin x}{x}dx$, so the Improper Riemann integral also exists.