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 that∫∞0|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)π∫nπ|sinxx|dx≥∞∑n=0(n+1)π∫nπ|sinx(n+1)π|dx=∞∑n=01(n+1)π(n+1)π∫nπ|sinx|dx=∞∑n=02(n+1)π=2π∞∑n=01n+1=2π(1+12+13+…)=∞
No comments:
Post a Comment