Write a value of 1 1+2e^ x dx - Vidyadip

Question

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

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Answer

Let $\int\frac{1}{1+2\text{e}^{\text{x}}}\text{dx}$
Dividing and multiplying by e^x
$\text{I}=\int\frac{\frac{1}{\text{e}^{\text{x}}}\text{dx}}{\frac{1}{\text{e}^{\text{x}}}+2}$
$=\int\frac{\text{e}^{-\text{x}}\text{dx}}{\text{e}^{-\text{x}}+2}$
Let $\text{e}^{-\text{x}}+2=\text{t}$
$-\text{e}^{-\text{x}}\text{dx}=\text{dt}$
$\text{e}^{-\text{x}}\text{dx}=-\text{dt}$
$\therefore\ \text{I}=-\int\frac{\text{dt}}{\text{t}}$
$=-\log|\text{t}|+\text{C}$
$=-\log|\text{e}^{-\text{x}}+2|+\text{C}$ $(\because\text{t}=\text{e}^{-\text{x}}+2)$

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