Friday, 22 March 2019

integration - Prove intinfty0frac|sinx|sinxxdx=1



Prove
0|sinx|sinxxdx=1.
I know how to calculate 0sinxxdx=π2, but the method cannot be applied here. So I am thinking
nk=0(1)k(k+1)πkπsin2xxdx
but I don't know how to proceed.


Answer



By Lobachevsky integral formula: https://en.wikipedia.org/wiki/Lobachevsky_integral_formula

0sinxx|sinx|dx=π/20|sinx|dx=1.


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