How do you evaluate the integral $int e^t/(e^t+1)dt$ ?

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Answer 1 · 2017-05-06T00:43:51

$int \ (e^t)/(e^t+1) \ dt = ln(e^t+1)+C$

We want to find:

$I = int \ (e^t)/(e^t+1) \ dt$

We can perform a simple substitution; Let

$u = e^t+1 => (du)/dt = e^t$

If we perform the substitution then we get:

$I=int color(red)(e^t)/u " " (du)/color(red)(e^t)$
$\ \ = int \ 1/u \ du$
$\ \ = ln|u| + C$

And restoring the substitution we get:

$I = ln(e^t+1)+C$

How do you evaluate the integral int e^t/(e^t+1)dtint e^t/(e^t+1)dt?