Integrate limit 0 to two pie (1 divided by 1 plus e to the power sin x) dx

Evaluate :

 \int\limits_0^{2\pi } {\frac{1}{{1 + {e^{\sin x}}}}dx}

Let {I = \int\limits_0^{2\pi } {\frac{1}{{1 + {e^{\sin x}}}}dx}  -  -  -  -  - (1)}

{ = \int\limits_0^{2\pi } {\frac{1}{{1 + {e^{ - \sin x}}}}dx} }   {\left\{ {\int\limits_0^a {f(x)dx = \int\limits_0^a {f(a - x)dx} } } \right\}}

{ = \int\limits_0^{2\pi } {\frac{{{e^{\sin x}}}}{{1 + {e^{\sin x}}}}dx}  -  -  -  -  -  - (2)}

Adding equations (1) and (2)

{2I = \int\limits_0^{2\pi } {1dx = 2\pi } }

{\therefore I = \pi }

 

 

 

 

 

 

 

Post Author: E-Maths