Friday, June 16, 2017

calculus - How to prove that intinfty0sin4(x)ln(x)cdotmathrmdxoverx2=piover4cdot(1gamma)?




How to prove that

0sin4(x)ln(x)dxx2=π4(1γ).




Here is my attempt:



I(a)=0ln(x)sin4(x)xadx



I(a)=0sin4(x)xadx



I(2)=0sin4(x)x2dx




I(2)=0sin2(x)x2dx140sin2(2x)x2dx=π2



Why is this way wrong?



How to prove (1)?


Answer



This is solved essentially in the same way explained in answers to your previous question. As a convenient starting point, I will refer to @Jack D'Aurizio's answer:



01cos(kx)x2log(x)dx=kπ2(1γlogk).




Now all you have to do is to write



sin4x=12(1cos(2x))+18(1cos(4x)).



I hope that the remaining computation is clear to you.






For your attempt, a correct computation would begin with




dda0sin4xxadx=0sin4xxalogxdx.



Notice that you misidentified the derivative of your parametrized integral.


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