[Q] square of 'sinc function' integral

[good_phy]

Hi.

I tried to solve some problem that i should get probability density with which eigenstate of

momentum is chosen after momentum measurement by using <\varphi_{k}|\Psi>

I faced some stuck integral problem such as \int_{k_o}^0\frac{sin^{2}(2x)}{2x^{2}}dx

I transformed sin^{2}(2x) = \frac{1 - cos(2x)}{2} so i obtained \int_{k_o}^0\frac{1-cos(2x)}{2x^{2}}dx but i don't know next step because, \int_{k_o}^0\frac{1}{x^2}dx go up to infinity,diverse.

i tried to do partial integral such as \int udv = uv - \int vdu but encountered same problem.

How can i overcome this singular point problem? i convinced that \int_{k_o}^0\frac{sin^{2}(2x)}{2x^{2}}dx

should be solved to convegence because graphic of \frac{sin^{2}(2x)}{2x^{2}}.

Please help me and give me an answer.

 

[jomoonrain]

i think you can try to write the cos as a series form.
P.S. the third formula in your statements seems...