Let $a>0$ be a constant.Consider the integral$$\int_0^{+\infty}\frac{x\cos(ux)}{a^2+x^2}dx.$$
Can I use Dirichlet's Test to proof that it's uniformly convergent on $[\delta,+\infty) (\forall \delta>0)$?
Obviously,$$\lvert\int_0^{+\infty}\cos(ux)dx\rvert\le2,\forall u\ge\delta.$$
However, the function$\frac{x}{a^2+x^2}$ is not monotonically decreasing on $(0,+\infty)$ but on $(a,+\infty)$.It's different to the theorem's condition.
That also makes me think the deep meaning of the "monotone" condition in this theorem.
For this integration, there are two parameters in total.Hence, there are two different understandings.Moreover,taking $u=\frac{1}{n} $ and $x=n$,we have $$ \left | \int_{0}^{n}\cos\left ( \frac{t}{n} \right ) dt\right |=n\sin(1) >2 $$ So,I can't understand $\left | \int_{0}^{+\infty }\cos\left ( ut\right ) dt\right |\le 2$
Case 1: If $a$ is fixed, we can consider the integration as $$ I(u)=\int_{0}^{+\infty} \frac{x \cos (u x)}{a^{2}+x^{2}} d x $$ Then we can always handle it in this way: $$ \int_{0}^{+\infty} \frac{x \cos (u x)}{a^{2}+x^{2}} d x =\int_{0}^{a} \frac{x \cos (u x)}{a^{2}+x^{2}} d x+\int_{a}^{\infty } \frac{x \cos (u x)}{a^{2}+x^{2}} d x $$ Note that $$ \left | \int _0^{x}\cos(ut)dt \right | =\left | \frac{\sin(ux)}{u} \right | \le \frac{1}{\delta } $$ Since the function$\frac{x}{a^2+x^2}$ is monotonically decreasing on $(a,+\infty)$,by Dirichlet's Test,$\int_{a}^{\infty } \frac{x \cos (u x)}{a^{2}+x^{2}} d x$ is uniformly convergent $\text { on }[\delta,+\infty)(\forall \delta>0)$.It follows that $$ \forall \varepsilon >0,\exists A>0,\forall A_1,A_2>A,\forall u\in[\delta,+\infty):\left | \int _{A_1}^{A_2}\frac{x \cos (u x)}{a^{2}+x^{2}}dx \right |<\varepsilon $$ This proves that $\int_{0}^{\infty } \frac{x \cos (u x)}{a^{2}+x^{2}} d x$ is uniformly convergent $\text { on }[\delta,+\infty)$.
Case 2: If $u$ is fixed,then we can consider the integration as $$ I(a)=\int_{0}^{+\infty} \frac{x \cos (u x)}{a^{2}+x^{2}} d x $$ This situation requires the use of some techniques to avoid the question you mentioned.Clearly,for all fixed $\delta>0$,we have $$ \int_{0}^{+\infty} \frac{x \cos (u x)}{\delta^{2}+x^{2}} d x<\infty \quad\forall \delta>0 $$ Note that $$ \begin{aligned} \int_{0}^{+\infty} \frac{x \cos (u x)}{a^2+x^{2}} d x&=\int_{0}^{+\infty} \frac{x \cos (u x)}{\delta ^2+x^{2}}\cdot \frac{\delta ^2+x^{2}}{a ^2+x^{2}} d x\\ &=\int_{0}^{+\infty} \frac{x \cos (u x)}{\delta ^2+x^{2}}\cdot \left ( 1+\frac{a^2-\delta ^2}{a ^2+x^{2}} \right ) d x \end{aligned} $$ Let $f(x)=1+\frac{a^2-\delta ^2}{a ^2+x^{2}}$.Since $f(x)$ is monotonic and $0<f(x)\le2$,by Abel's test,$\int_{0}^{\infty } \frac{x \cos (u x)}{a^{2}+x^{2}} d x$ is uniformly convergent $\text { on }[\delta,+\infty)$.