Question

The value of $\underset{0}{\overset{\pi }{\int }}\frac{e^{\cos x}}{e^{\cos x}+e^{-\cos x}}dx$ is

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

Answer 1

Answer: (D)

$I=\underset{0}{\overset{\pi }{\int }}\frac{e^{\cos x}}{e^{\cos x}+e^{-\cos x}}dx=\underset{0}{\overset{\pi }{\int }}\frac{e^{\cos (\pi -x)}}{e^{\cos (\pi -x)}+e^{-\cos (\pi -x)}}dx$
$=\underset{0}{\overset{\pi }{\int }}\frac{e^{-\cos x}}{e^{-\cos x}+e^{\cos x}}dx$
$\therefore 2I=\underset{0}{\overset{\pi }{\int }}\frac{e^{\cos x}+e^{-\cos x}}{e^{\cos x}+e^{-\cos x}}dx=\underset{0}{\overset{\pi }{\int }}dx=\pi$
$I=\frac{\pi }{2}$