I am looking for a short proof that ∫∞0(sinxx)2dx=π2.
What do you think?
It is kind of amazing that ∫∞0sinxxdx is also π2. Many proofs of this latter one are already in this post.
Well, it’s not hard to reduce this integral to ∫∞0sin(x)xdx: Just integrate by parts in ∫∞0sin2(x)x2dx, integrating 1x2 and differentiating sin2(x). You’re left with ∫∞0sin(2x)xdx which reduces to the ∫∞0sin(x)xdx integral after changing variables from x to 2x.
So any elementary proof that ∫∞0sin(x)xdx=π2 is effectively also an elementary proof that ∫∞0sin2(x)x2dx is also π2.