Consider the integral $(1)$
$$\text{Conjecture}\ \int_{-\infty}^{+\infty}{\mathrm dx\over x^2}\left(1-{\sin x\over x}\right)^n=\color{blue}{\pi\over n+1}\tag1$$ Where $n\ge1$
Required help to prove conjecture $(1)$
An attempt:
Apply binomial series to $(1)$, then we have
$$\sum_{k=0}^{n}(-1)^k{n\choose k}\int_{-\infty}^{+\infty}{x^{n-k}\sin^k x\over x^2}\mathrm dx\tag2$$
We can apply the trick of $(3)$
$$\int_{-\infty}^{+\infty}f(x){\sin^2 x\over x^2}\mathrm dx=\int_{0}^{\pi}f(x)\mathrm dx\tag3$$ then $(2)$ becomes $$I_k=\sum_{k=0}^{n}(-1)^k{n\choose k}\int_{0}^{\pi}{x^{n-k}\sin^{k-2} x}\mathrm dx\tag4$$
I guess we could apply integration by parts to $(4)$ but seem difficult...