Thursday, 13 February 2014

real analysis - Calculate the integral intinfty0|fracsinxx|dx



I'm trying to solve this problem from Real Analysis of Folland but can't find any solution for it. Can anyone help me ?. Thanks so much.



Show that0|sin(x)x|dx=




And also, can we calculate the similar integral
0sin(x)xdx ?. Please help me clarify this. I really appreciate.


Answer



0|sinxx|dx=n=0(n+1)πnπ|sinxx|dxn=0(n+1)πnπ|sinx(n+1)π|dx=n=01(n+1)π(n+1)πnπ|sinx|dx=n=02(n+1)π=2πn=01n+1=2π(1+12+13+)=


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