Monday, 27 August 2018

calculus - Convergence of the integral inti0nftyfracsin2xx2 mathrmdx.





Determine whether the integral 0sin2xx2 dx converges.




I know it converges, since in general we can use complex analysis, but I'd like to know if there is a simpler method that doesn't involve complex numbers. But I cannot come up with a function that I could compare the integral with.


Answer



Hint:$$x>1\implies0\le\frac{\sin^2(x)}{x^2}\le\frac1{x^2}\\0

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