Let \(g(t)={\int }_{-\pi /2}^{\pi /2}\cos \left(\frac{\pi }{4}t+f(x)\right)dx\), where \(f(x)={\log }_{e}\left(x+\sqrt{{…

Let \(g(t)={\int }_{-\pi /2}^{\pi /2}\cos \left(\frac{\pi }{4}t+f(x)\right)dx\), where \(f(x)={\log }_{e}\left(x+\sqrt{{x}^{2}+1}\right),x\in R\). Then which one of the following is correct?

[JEE Main 2021, 20 Jul (Shift 2)]