Thursday, 11 April 2019

Test for divergence of intinfty0fracsin2(x)xdx without evaluating the integral



I would like to prove that 0sin2(x)xdx diverges without actually evaluating the integral. Is there a convergence test from calculus or real analysis that can show that this integral diverges?



Thanks.



Edit: Someone pointed out that this is a possible duplicate. However, the question put forth as a possible duplicate asks about sin(x2), not about sin2(x).


Answer



It is a divergent integral by Kronecker's lemma, since sin2(x) is a non-negative function with mean value 12. In more explicit terms, by integration by parts we have




Nππsin2(x)xdx=[12sin(2x)4x]Nππ+12Nππdxx+O(1)
where the blue terms are bounded, but the red term equals 12logN.


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