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integrate 1/(1+e^sinx) from 0 to 2pie

integrate 1/(1+e^sinx) from 0 to 2pie
 

Grade:12

1 Answers

Latika Leekha
askIITians Faculty 165 Points
9 years ago
Hello student,
The given integral is
\int_{0}^{2\pi }\frac{dx}{1+e^{sin x}}
Let I = \int_{0}^{2\pi }\frac{dx}{1+e^{sin x}} ... (1)
We know \int_{0}^{a}f(x) dx = \int_{0}^{a}f(a-x)dx
So I = \int_{0}^{2\pi }\frac{dx}{1+e^{-sin x}}
I = \int_{0}^{2\pi }\frac{e^{sin x}dx}{1+e^{sin x}} .... (2)
Adding (1) and (2) we get
2I = \int_{0}^{2\pi }\frac{1+e^{sin x}}{1+e^{sin x}}dx
2I = 2π
So I = π.

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