Q:

Integral involving sgn function

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. 

A:

$$ \DeclareMathOperator{\sgn}{\operatorname{sgn}} \sgn(t-1)- \sgn(t+1)=
\begin{cases} 
0, &\text{if } |t| \geq 1, \\
-2, &\text {if } |t| < 1 .  
\end{cases}
$$
$$\begin{align*}\int_{-\infty}^{\infty}(t^2-1)^{pn}(\sgn(t-1)- \sgn(t+1))^p \mathrm dt &=\int_{-1}^{1}(t^2-1)^{pn}(-2)^p \mathrm d t\end{align*}$$
Hint:
Essentially, the problem involves breaking up the integral into relevant pieces and looking the signum function in those pieces.  For instance, when $t>1$, note that the integral vanishes and so on.