Let \( g(x) \) be a real valued function defined on the interval \(... VIDEO
Let \( g(x) \) be a real valued function defined on the interval \( \left(\frac{-\pi}{2}, \frac{\pi}{2}\right) \) such that
\( \mathrm{P} \) \( g(x)=e^{2 x}+\int_{0}^{\sin x} \frac{e^{t}}{\cos ^{2} x+2 t \sin x-t^{2}} d t \forall x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right) \). Also \( f(x) \) be the inverse
W function of \( g(x) \), where \( 0 \leq x \leq \frac{\pi}{2} \). Find the value of \( \frac{1}{\left(f^{\prime}(1)\right)^{2}}+g(0)+g^{\prime}(0)+g^{\prime \prime}(0) \).
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