Evaluate the following functions : 1 1+e^ -x d x - Vidyadip

Question

Evaluate the following functions : $\int \frac{1}{1+e^{-x}} \cdot d x$

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Answer

$I=\int \frac{1}{1+e^{-x}} \cdot d x$
$
\begin{aligned}
& =\int \frac{1}{1+\frac{1}{e^x}} \cdot d x \\
& =\int \frac{1}{\frac{e^x+1}{e^x}} \cdot d x \\
& =\int \frac{e^x}{e^x+1} \cdot d x \\
\because \quad & \frac{d}{d x}\left(e^x+1\right) \cdot d x=e^x \\
& =\log \left[e^x+1\right]+c
\end{aligned}
$

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