Why does $\int_1^\infty\frac{\sin^2(x)}{x}\mathrm d x$ diverge?

Why does $\displaystyle\int_1^\infty\dfrac{\sin^2(x)}{x}$ diverge ?

Why does Dirichlet-Test not work ?

Define $f(x)=g(x)=\dfrac{\sin(x)}{x^{1/2}}$, Then $\forall b>1$ and $a>0;$ $$\Big\lvert\displaystyle\int_1^b\dfrac{\sin(x)}{x^{a}}\Big\rvert<+\infty$$

also for $a=1/2$

and since $\lim\limits_{x\to\infty}\dfrac{\sin(x)}{x^{1/2}}=0$

The integral should be convergent, did I overlook something ?

but with the same method of Dirichlet, one can show the divergence

$\displaystyle\int\dfrac{\sin^2(x)}{x}=\displaystyle\int\dfrac{1-\cos(2x)}{x}=\underbrace{\displaystyle\int\dfrac{1}{x}}_{divergent}-\underbrace{\displaystyle\int\dfrac{\cos(2x)}{x}}_{convergent}$

-
$\sin$ changes signs, $\sin^2$ does not. –  Thomas Apr 20 at 12:51
@Thomas but in the statement I couldn't find anything about the signs. –  derivative Apr 20 at 12:52
For Dirichlet's criterion, you need one factor with bounded integrals, and one monotonic. You have no monotonic factor here, when you use $$\frac{\sin x}{\sqrt{x}}\cdot \frac{\sin x}{\sqrt{x}}.$$ If you take a different decomposition with a monotonic factor, the integrals of the other aren't bounded. –  Daniel Fischer Apr 20 at 12:52
@Daniel Fischer Yes that was the point. –  derivative Apr 20 at 12:55

For example because, for every $n\geqslant1$, $$\int_{n\pi+\pi/4}^{n\pi+3\pi/4}\frac{\sin^2x}x\mathrm dx\geqslant\int_{n\pi+\pi/4}^{n\pi+3\pi/4}\frac{\frac12}{(n+1)\pi}\mathrm dx=\frac1{4(n+1)}$$