Mathematics

Write a value of 
$$\displaystyle \int { \cfrac { 1 }{ 1+2{ e }^{ x } }  } dx$$


SOLUTION
$$I=\displaystyle\int{\dfrac{1}{1+2{e}^{x}}dx}$$

$$=\displaystyle\int{\dfrac{{e}^{-x}}{{e}^{-x}\left(1+2{e}^{x}\right)}dx}$$

$$=\displaystyle\int{\dfrac{{e}^{-x}}{\left(2+{e}^{-x}\right)}dx}$$

Let $$t=2+{e}^{-x}\Rightarrow\,dt=-{e}^{-x}dx$$

$$=\displaystyle\int{\dfrac{-dt}{t}}$$

$$=-\log{\left(t\right)}+c$$        .....where $$c$$ is the constant of integration.

$$=-\log{\left(2+{e}^{-x}\right)}+c$$     ...........where $$t=2+{e}^{-x}$$
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Subjective Medium Published on 17th 09, 2020
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