We have the following test prep question, for a measure theory course:
$\forall s\geq 0$, define $F(s)=\int_0^\infty \frac{\sin(x)}{x}e^{-sx}\ dx.$
a) Show that, for $s>0$, $F$ is differentiable and find explicitly its derivative.
b) Keeping in mind that $F(s)=\int_0^\pi \frac{\sin(x)}{x}e^{-sx}\ \ dx\ +\int_\pi^\infty \frac{\sin(x)}{x}e^{-sx}\ dx,$ and conveniently doing integration by parts on the second integral on the right hand side of the previous equation, show that $F(s)$ is continuous at $s=0$. Calculate $F(s)\ (s\geq 0)$.
Since it's a measure theory course, I'm thinking there are methods involving the things you typically learn in these courses, and I think Lebesgue's Dominated Convergence Theorem will play a role, because I was looking at books by Bartle and Apostol, and they both have similar exercises or theorems, and both use LDCT.
Also, I suppose these proofs regarding continuity or differentiability could be done with standard calculus stuff (like $\epsilon$'s and $\delta$'s or the actual definition of a derivative), but I want to avoid these methods and focus on what I should be learning from the class.
I think I have part (a), or at least a good idea, based on the Bartle book. If I let $f(x,s)=\frac{\sin(x)}{x}e^{-sx}$, I just need to find an integrable function $g$ such that $\big|\frac{\partial f}{\partial s}\big|\leq g(x)$ (after showing that partial does exist, of course :) ). And then, $\frac d{ds}F(s)=\int _{\mathbb{R}^+}\frac{\partial f}{\partial s}\ dx.$ Please correct me if I'm mistaken, or missing something.
Now, for part (b) I'm a little stumped. In the Apostol book, the case $s>0$ is done explicitly, but I read through it and it didn't help me. Looking at the Bartle book, I get the idea of defining $f_n=(x,s_n)$, where $s_n=\frac1{n+1}$ or some such sequence that goes to zero. Then, somehow, maybe, LDCT kicks in (but I guess I'd have to find a function what would dominate these $f_n$). I also don't really see the point in dividing the integral into the two parts up there, so I must be missing something.