Question

Evaluate each of the following integral:

\[\int_0^{2\pi} \frac{e^\ sin x}{e^\ sin x + e^{- \ sin x}}dx\]

 

Answer 1

\[\text{Let I }= \int_0^{2\pi} \frac{e^\ sin x}{e^\ sin x + e^{- \ sin x}}dx\]          ....................(1)

Then,

\[I = \int_0^{2\pi} \frac{e^\ sin\left( 2\pi - x \right)}{e^\ sin\left( 2\pi - x \right) + e^{- \ sin \left( 2\pi - x \right)}}dx .....................\left( \int_0^a f\left( x \right)dx = \int_0^a f\left( a - x \right)dx \right)\]
\[ = \int_0^{2\pi} \frac{e^{- \ sin x}}{e^{- \ sin x} + e^\ sin x}dx ..........................\left( 2 \right)\]

Adding (1) and (2), we get

\[2I = \int_0^{2\pi} \frac{e^\ sin x + e^{- \ sin x}}{e^\ sin x + e^{- \ sin x}}dx\]
\[ \Rightarrow 2I = \int_0^{2\pi} dx\]
\[ \Rightarrow 2I = x_0^{2\pi} \]
\[ \Rightarrow 2I = 2\pi - 0\]
\[ \Rightarrow I = \pi\]