Question

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

• A. $g(1)=g(0)$
• B. $\sqrt{2} g(1)=g(0)$
• C. $g(1)=\sqrt{2} g(0)$
• D. $g(1)+g(0)=0$

Answer 1

Answer: (B)

$g(t)=\int\limits_{-\pi / 2}^{\pi / 2}\left(\cos \frac{\pi}{4} t+f(x)\right) d x$ $g(t)=\pi \cos \frac{\pi}{4} t+\int\limits_{-\pi / 2}^{\pi / 2} f(x) d x$ $g(t)=\pi \cos \frac{\pi}{4} t$ $g(1)=\frac{\pi}{\sqrt{2}}, g(0)= pi$