To show a function is integrable

Question

I came across this question while studying for my exam: To show the function $f(x) = \frac{(\sin x)^2}{x^2}$ is Lebesgue integrable on [0, $\infty$). I wonder if there is any smarter way of proving it (like, using absolute continuity) without going all the way back to the definition. Thanks

Answer 1

A standard way to show that a "complicated" non-negative function is integrable is to find an integrable upper bounding function (for which integrability is simpler to prove).

Since for all $t$, $|\sin t|\leqslant \min\{|t|,1\}$, we get for $x$ positive, $$0\leqslant \frac{\sin^2x}{x^2}\leqslant \chi_{(0,1]}+\chi_{(1,+\infty)}\frac 1{x^2}.$$