How do you find the antiderivative of $(e^(2x))/(1+e^(2x))$ ?

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Answer 1 · 2017-02-22T14:27:10

$int \ e^(2x)/(1+e^(2x)) \ dx = 1/2ln|1+e^(2x)| + c$

We want to find $int \ e^(2x)/(1+e^(2x)) \ dx$

A trivial substitution can be used to simplify the denominator; Let:

$u = 1+e^(2x)$

Then $(du)/dx = 2e^(2x) => 1/2 (du)/dx = e^(2x)$

If we substitute this into the integral we get;

$int \ e^(2x)/(1+e^(2x)) \ dx = int \ (1/2)/(u) \ du$
$" " = 1/2ln|u| + c$

And if we undo the substitution we get:

$int \ e^(2x)/(1+e^(2x)) \ dx = 1/2ln|1+e^(2x)| + c$

How do you find the antiderivative of (e^(2x))/(1+e^(2x))(e^(2x))/(1+e^(2x))?