Let \( f: R \rightarrow\left[\frac{3}{4}, \infty\right) \) be a surjective quadratic function wi...

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Let \( f: R \rightarrow\left[\frac{3}{4}, \infty\right) \) be a surjective quadratic function with line of symmetry \( 2 x-1=0 \) and \( f(1)=1 \)
If \( g(x)=\frac{f(x)+f(-x)}{2} \) then \( \int \frac{d x}{\sqrt{g\left(e^{x}\right)-2}} \) is equal to :
(a) \( \sec ^{-1}\left(e^{-x}\right)+C \)
(b) \( \sec ^{-1}\left(e^{x}\right)+C \)
(c) \( \sin ^{-1}\left(e^{-x}\right)+C \)
(d) \( \sin ^{-1}\left(e^{x}\right)+C \)
(Where \( C \) is constant of integration)
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