Let \(g(t)=\int_{\frac{-\pi} { 2}}^{\frac{\pi} { 2}} \cos \left(\frac{\pi}{4} t+f(x)\right) d x, \) where \(f(x)=\log _e…

Let \(g(t)=\int_{\frac{-\pi} { 2}}^{\frac{\pi} { 2}} \cos \left(\frac{\pi}{4} t+f(x)\right) d x, \) 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)]