Question

Integrate ${\int }_0^{\pi }\frac{e^{\cos x}}{e^{\cos x}+e^{-\cos x}}dx$

• A. $\frac{\pi }{12}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is D $\frac{\pi }{2}$

$I={\int }_0^{\pi }\frac{e^{\cos x}}{e^{\cos x}+e^{-\cos x}}dx$ ...(1)
$\Rightarrow I={\int }_0^{\pi }\frac{e^{\cos (\pi -x)}}{e^{\cos (\pi -x)}+e^{-\cos (\pi -x)}}$

$\Rightarrow {\int }_0^{\pi }\frac{e^{-\cos x}}{e^{-\cos x}+e^{\cos x}}dx$ ...(2)
Adding (1) and (2)
$2I={\int }_0^{\pi }dx=\pi$

$\Rightarrow I=\frac{\pi }{2}$