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#