[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...