Mathematics

The value of $$\displaystyle \int_{0}^{2\pi}\frac{dx}{e^{\sin x}+1}$$


ANSWER

$$\displaystyle \pi $$


SOLUTION
Let $$\displaystyle I=\int_{0}^{2\pi}\frac{dx}{e^{\sin x}+1}$$...(1)

$$\displaystyle I=\int _{ 0 }^{ 2\pi  } \frac { dx }{ e^{ -\sin  x }+1 } $$

$$\displaystyle  I=\int _{ 0 }^{ 2\pi  } \frac { e^{ \sin  x } }{ e^{ \sin  x }+1 } $$...(2)

$$\displaystyle 2I=\int _{ 0 }^{ 2\pi  } \frac { e^{ \sin  x }+1 }{ e^{ \sin  x }+1 } dx$$

$$\displaystyle 2I=2 \pi $$

$$\Rightarrow I=\pi $$
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Single Correct Medium Published on 17th 09, 2020
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