show that int(sin^2(x)/ (x^2))dx from 0 to infinity converges and THEN use integration by parts and the identity sin(2x)=2sin(x)cos(x) to show that
int(sin^2(x)/ (x^2))dx from 0 to infinity = int(sin(x) / x) dx
You can see here : http://www.mathhelpforum.com/math-he...-integral.html for something similar, proving it's convergent.
(similar method as (1-cos)/x²)
As for the integration by parts, integrate 1/x² and differentiate sin²(x) (which will give 2sin(x)cos(x), and this is where you'll use the identity)
And this would rather be int(sin(2x)/x), wouldn't it ?