Friday, 9 November 2018

calculus - Show that intinfty0left(fracsin(x)xright)2<2



I`m trying to show that this integral is converges and <2
0(sin(x)x)2dx<2
What I did is to show this expression:
10(sin(x)x)2dx+1(sin(x)x)2dx
Second expression :
1(sin(x)x)2dx<1(1x2)2dx=lim
Now for the first expression I need to find any explanation why its <1 and I will prove it.

I would like to get some advice for the first expression. thanks!



Answer



Hint: \lim_{x\to0}\frac{\sin x}{x}=1.


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