JEE Math Practice Question

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?

1. A.g(1) = g(0)
2. B.$$\sqrt 2 g(1) = g(0)$$
3. C.$$g(1) = \sqrt 2 g(0)$$
4. D.g(1) + g(0) = 0