Question

${\int }_0^{\pi }\frac{dx}{1+a^{\cos x}}$ equals

• A. $0$
• B. $\frac{\pi }{2}$
• C. $\pi$
• D. None of these

Answer 1

Answer: (B)

$\frac{\pi }{2}$

$I={\int }_0^{\pi }\frac{dx}{1+a^{\cos x}}={\int }_0^{\pi }\frac{dx}{1+a^{\cos (\pi -x)}}={\int }_0^{\pi }\frac{dx}{1+a^{-\cos x}}={\int }_0^{\pi }(1-\frac{1}{1+a^{\cos x}})dx$
$\Rightarrow I=\pi -I$
$\Rightarrow I=\frac{\pi }{2}$
Ans: B