Disguised as Integral

Calculus Level 5

The integral \(\displaystyle I = \int \limits_0^{\frac{\pi }{2}} \frac{ \text{d}x }{(1 + \sin x )(1 + \sin x - \cos x )} \) can be expressed as an infinite sum i.e., \(\displaystyle \lim_{n \to \infty} S(n) \). It is found that the limit does not exist.

However, the limit \( \displaystyle L = \lim_{ n \to \infty } \frac{e^{S(n)}}{n} \) exists.

Evaluate: \( \text{ln } L \).

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