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Question
\[\int\frac{e^x}{\left( 1 + e^x \right)^2} dx\]
Sum
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Solution
\[\int\frac{e^x dx}{\left( 1 + e^x \right)^2}\]
\[\text{Let 1 }+ e^x = t\]
\[ \Rightarrow e^x = \frac{dt}{dx}\]
\[ \Rightarrow e^x dx = dt\]
\[Now, \int\frac{e^x dx}{\left( 1 + e^x \right)^2}\]
\[ = \int\frac{dt}{t^2}\]
\[ = \int t^{- 2} dt\]
\[ = \frac{t^{- 2} + 1}{- 2 + 1} + C\]
\[ = \frac{- 1}{t} + C\]
\[ = - \frac{1}{1 + e^x} + C\]
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