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 #