Proof of ∫∞0(sinxx)2dx=π2.\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.

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.

Answer

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.

Attribution
Source : Link , Question Author : TCL , Answer Author : Aryabhata

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