An integral limit!

Calculus Level 5

\[\displaystyle {\left(\lim_{\substack{n\rightarrow \infty \\ n \in \mathbb{Z}}} \frac{\displaystyle \int_{0}^{\pi/2}{\left(\sin(x) + \cos(x)\right)^{n+1}\ dx}}{\displaystyle \int_{0}^{\pi/2}{\left(\sin(x) + \cos(x)\right)^n\ dx}}\right)}^2 = \, ?\]

This is a limit of sequences

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