(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

let f:[0,oo) -> R be given by f(x) = sin(x) / x for x>0 and f(0) = c. Prove that f is improper riemann integrable without computing the integral explicitly

3. The attempt at a solution

I've attempted to find a upperbound for f(x) such that the integral does not diverge. The most simple one is to use the fact that sin(x) <= 1 for all x, but this gives a divergent integral.

I've already proved that f is Lebesgue measurable for every c in R. So I could turn the integral [tex]\int_0^R f(x) dx[/tex] into a Lebesgue integral and then use one of the convergence theorems to try to show with them that the integral does not diverge. But I haven't succeded in doing this :(

Can anyone help me out?

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# Proving function is improper riemann integrable

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