General term of a sequence $(2-1)(2+1)(3-1)(3+1)...(n-1)(n+1)$ Can we use integrals, and are there some general methods for finding terms of a sequence?

Let's reorder the terms and recognize the result:
$$
\begin{align}P&=(2-1)(2+1)(3-1)(3+1)\dots(n-1)(n+1)
\\&=(2-1)(3-1)\dots(n-1)\times(2+1)(3+1)\dots(n+1)
\\&=\frac{(n-1)!(n+1)!}{2}
\end{align}
$$
