Evaluate the following integrals: e^x (1+e^x)^2 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{\text{e}^\text{x}}{(1+\text{e}^\text{x})^2}\text{dx}$

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Answer

$\int\frac{\text{e}^\text{x}}{(1+\text{e}^\text{x})^2}\text{dx}$
$\text{Let}\ 1+\text{e}^\text{x}=\text{t}$
$\Rightarrow\text{e}^\text{x}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\text{e}^\text{x}\text{dx}=\text{dt}$
$\text{Now},\int\frac{\text{e}^\text{x}\text{dx}}{(1+\text{e}^\text{x})^2}$
$=\int\frac{\text{dt}}{\text{t}^2}$
$=\int\text{t}^{-2}\text{dt}$
$=\frac{\text{t}^{-2}+1}{-2+1}+\text{C}$
$=\frac{-1}{\text{t}}+\text{C}$
$=-\frac{1}{1+\text{e}^\text{x}}+\text{C}$

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