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6 maj 2012 · First we split $\sin^2(x)=\frac{(1-e^{2ix})+(1-e^{-2ix})}{4}$. To avoid the pole at $x=0$, drop the path of integration a bit below the real line (this function has no poles and it vanishes at infinity, so this is okay).
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I have to show that $f(x)=\sin(x^2)$ is integrable on $[1, \infty[$. This is French terminology, so "intégrable" specifically means that the integral of $|f|$ exists. The only method I know is to compare it to functions of the form $\frac{1}{x^\alpha}$, but it's not eventually smaller or larger than any of these.
