I find it hard to find an appropriate dominating function for the integral
$$I:=\lim_{t \to \infty}\int_{(0, \infty)} \frac{1-e^{-tx}(x \ sin t + \cos t)}{1+x^2}d \lambda_1(x), \ t > 0 $$
Indeed if I set for $t>0$ $$f_t(x) = \frac{1-e^{-tx}(x \sin t + \cos t)}{1+x^2} $$ then $f_t \to 1/(1+x^2) $ point wise as $t \to \infty$ and changing the limit with the integral it would follow that $I= \pi /2 $
However to apply dominated convergence I need to dominate $|f_t|$ by an integrable function $g$ and I find it hard to come up with such a function. It is easy to establish that $$|f_t(y)| \leq \frac{2+x}{1+x^2} $$ but said upper bound is not integrable on $(0, \infty)$.
I can also control the values of $x$ by splitting the integral into $(0,1)$ and $(1, \infty)$ but I don't see how I can do such with my $t>0$, I suppose that my dominating function $g$ has to be independent of $t$ and I cannot use arguments such as:
- For large enough $t>0$ consider that $xe^{-tx} \leq 1$
I also thought about using the idea of an integrable function $g$ that dominates $f_t$ almost everywhere, but couldn't come up with such a function either, in particular showing that the set where it wouldn't dominate $f_t$ is a null set.