Solving simple problems 3 - Dieudonne 'Infinitesimal Calculus'

This is not really a simple problem, but still fun.

In the neighborhood of $+\infty$, the function $\frac{\sin x}{\sqrt{x}} + \frac{\sin^2 x}{x}$ has for generalized principal part the function $\frac{\sin x}{\sqrt x}$; but the integral $\int^{+\infty}_1 \frac{\sin t}{\sqrt t} \ dt$ is convergent whereas the integral $\int_{1}^{+\infty} \left( \frac{\sin t}{\sqrt t} + \frac{\sin^2 t}{t} \right) \ dt$ is not convergent.

Solution 

$\int_{1}^{+\infty} \left( \frac{\sin t}{\sqrt t} + \frac{\sin^2 t}{t} \right) \ dt < 0$

$\epsilon > \frac{1}{2} + 1$

$\delta = 1 + \frac{1}{2}$

$ \frac{\sin x}{\sqrt x} > \frac{1}{5} + \frac{1}{2} + 1$