Direct comparison test condition

Question

I tried to use direct comparison test to check whether it converges or diverges, I always confuse how to find the inequalities

1. $\int_0^\pi\frac{dt} {\sqrt t+sin x}\, \mathrm dx,$
for
$0\le t \le\pi$, $0 \le \frac {1}{\sqrt t +sin t} \le \frac{1}{\sqrt t}$ and

2. $\int_\pi^\infty\frac{1+sin x} {x^2}\, \mathrm dx,$

$0\le \frac{1+sinx}{x^2} \le \frac{2}{x^2}$, for $x \geq \pi$
is this because $0 \le sinx \le 1$?

can someone explain to me how can I get this condition?