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Integral Calculus> integrate 1/(1+e^sinx) from 0 to 2pie...

question mark

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

Karika Batra , 10 Years ago

Grade 12

1 Answers

Latika Leekha

Last Activity: 10 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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