Let $\mathrm{g}(\mathrm{t})=\int_{-\pi / 2}^{\pi / 2} \cos \left(\frac{\pi}{4} \mathrm{t}+f(\mathrm{x})\right) \mathrm{dx}$, where
$f(x)=\log _{e}\left(x+\sqrt{x^{2}+1}\right), x \in \mathbf{R}$. Then which one of the following is correct?
Correct Option: , 2
$\mathrm{g}(\mathrm{t})=\int_{-\pi / 2}^{\pi / 2}\left(\cos \frac{\pi}{4} \mathrm{t}+\mathrm{f}(\mathrm{x})\right) \mathrm{d} \mathrm{x}$
$\mathrm{g}(\mathrm{t})=\pi \cos \frac{\pi}{4} \mathrm{t}+\int_{-\pi / 2}^{\pi / 2} \mathrm{f}(\mathrm{x}) \mathrm{dx}$
$\mathrm{g}(\mathrm{t})=\pi \cos \frac{\pi}{4} \mathrm{t}$
$\mathrm{g}(1)=\frac{\pi}{\sqrt{2}}, \mathrm{~g}(0)=\pi$
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