I am having trouble with calculating or approximation the following integral: $ \int_{-\infty}^{\infty}(t^2-1)^{pn}(\mathrm{sgn}(t-1)-\mathrm{sgn}(t+1))^pdt, $ where $1\leq p<\infty$ and $n \in N$.
Any ideas or references would be very helpful. Thank you.