Please don't overthink

Algebra Level 4

If \[ f(x) = \left| \begin{array}{ccc} \\sin^5 x & \ln(\sin x) & \frac{\sqrt{\sin x}}{ \sqrt{\sin x } + \sqrt{\cos x}} \\ \pi^{ln(2)} & \displaystyle \lim_{n\to\infty} \displaystyle \sum_{k=1}^{n} \sin {\frac{2k\pi}{n+1}} \mp i\cos {\frac{2k\pi}{n+1}} & \prod_{k=1}^{n} k \\ \frac{8}{15} & \frac{\pi}{2} \ln{\frac{1}{2}} & \frac{\pi}{4} \end{array} \right|.\]

,then the value of \( \int\limits_{0}^{\frac{\pi}{2}} f(x) dx \)

Here is n is any natural number.

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