Definite Integration

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