I am looking at a proof and got stuck on a part with an integral. I tried to simplify the problem as much as possible, I hope I did not omit any potential helpful information. I have an even function $K()$ and I taking the integral of it and the product of a $\sin()$ squared. What I am confused about is that $K(0)$ was pulled out of the integral in one of the steps. Here is the integral,
$$\left ( \int_{-\infty}^\infty \frac{K(x)}{x}sin(x) dx\right )^2 $$
Then on the next line of the proof I see:
$$ K^2(0)\left (\int_{-\infty}^\infty \frac{1}{x}sin(x) dx\right )^2 (1+ o(1))$$
And then the proof continues. I am not sure while we are able to take the $K()$ function out. My suspicion is it has to do with some property about even and odd functions? Suggestions and help are appreciated!