Suppose that $f$ is a smooth function which satisfies in the following condition

$$f(\pi+x)=f(x)\space    \space  and   \space   f(\pi-x)=f(x)$$ then, if the following integral exists 

$$\int_0^\infty\frac{\sin^2x}{x^2}f(x)$$
We have the following nice equalities


$$\int_0^\infty\frac{\sin^2x}{x^2}f(x)=\int_0^{\infty}\frac{\sin x}{x}f(x)=\int_0^{\pi/2}f(x)dx$$

I proved the following formula as a part of my Bachelor project that 
 if $f(\pi+x)=f(x)\space    \space  and   \space   f(\pi-x)=f(x)$ then 

$$\int_0^\infty\frac{\sin^4x}{x^4}f(x)=\int_0^{\pi/2}f(x)dx+\frac{2}{3}\int_0^{\pi/2}\sin^2 x\,f(x)dx$$

and I gave a method for finding an explicit formula for higher degree

$$\int_0^\infty\frac{\sin^{2n}x}{x^{2n}}f(x)$$

see http://arxiv.org/pdf/1004.2653.pdf