Suppose that the function \( f, g, f^{\prime} \) and \( g^{\prime} \) are continuous over \( [0,...

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Suppose that the function \( f, g, f^{\prime} \) and \( g^{\prime} \) are continuous over \( [0,1], g(x) \neq 0 \) for \( x \in[0,1], f(0)=0, g(0)=e f(1)=\frac{2023}{2} \) and \( g(1)=1 \). Find the value of the definite integral, \( \int_{0}^{1} \frac{f(x) \cdot g^{\prime}(x)\left\{g^{2}(x)-1\right\}+f^{\prime}(x) \cdot g(x)\left\{g^{2}(x)+1\right\}}{g^{2}(x)} d x \).
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