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AllRelative

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## Homework Statement

(FYI It's from an Real Analysis class.)

Show that $$\int_{0}^{\infty} (sin^2(t) / t^2) dt $$ is convergent.

## Homework Equations

I know that for an integral to be convergent, it means that :

$$\lim_{x\to\infty} \int_{0}^{x} (sin^2(t) / t^2) dt$$ is finite.I can also use the fact that let: $$ f(x) = sin^2(t) / t^2 $$

and

Let :$$F(x) = \int_{0}^{x} (sin^2(t) / t^2) dt$$

Since f(x) is always positive from 0 to infinity. If F(x) has an upper limit that is not infinite, than the integral in convergent.

I've also seen a few other concepts around those like absolute convergence, Cauchy's criteria.

## The Attempt at a Solution

What I have been able to do is using Chasle relation, I proves it's convergence from 1 to infinity since for values of t from 1 to infinity

$$sin^2(t) / t^2 \leqslant 1 / t^2$$.

And knowing that $$\int_{1}^{\infty} 1 / t^2 dt$$ converges, then we know that $$ \int_{1}^{\infty} (sin^2(t) / t^2) dt$$ converges also.I have no clue on how to attack values of t from 0 to 1.

Thanks for the help.