∫ e ^ x 1+ e ^ x #160; dx e ^ x =m e ^ x dx=dm ⇒∫ 1 m ^ 2 ( m+1 ) #160; #160; dm =∫[ 1 m+1 1 m + 1 m ^ 2 #160; ...

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Integration by Substitution

Evaluate ∫e...

Question

Evaluate $\int \frac{e^{- x}}{1 + e^{x}} d x$

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Solution

$\int \frac{e^{- x}}{1 + e^{x}} d x$
$e^{x} = m$
$e^{x} d x = d m$
$\Rightarrow \int \frac{1}{m^{2} (m + 1)} d m$
$= \int [\frac{1}{m + 1} - \frac{1}{m} + \frac{1}{m^{2}}]$
$\Rightarrow I n | m + 1 | - I n | m | - \frac{1}{m} + c$
$\Rightarrow I n | e^{x} + 1 | - I n e^{x} - e^{- x} + c$
$= I n | e^{x} + 1 | - x - e^{- x} + c$

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