U-substitution for integral of 1/(1+e^x)dx. What am I doing wrong? Here is my work, with the right

Damon Stokes 2022-06-30 Answered
U-substitution for integral of 1/(1+e^x)dx. What am I doing wrong?
Here is my work, with the right answer. I feel like every step is right, but somehow I am getting the wrong answer. How?
1 1 + e z d z = 1 e z ( 1 e z + 1 ) d z = 1 1 e z + 1 1 e z d z
subbing u = 1 e z + 1, d u d z = e z d z d u = 1 e z d z
1 1 + e z d z = 1 u d u = ln ( u ) + C = ln ( e z + 1 ) + C
But the right answer is z ln ( 1 + e z ) + C
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Answers (2)

Arcatuert3u
Answered 2022-07-01 Author has 30 answers
Your answer is correct.
ln ( e z + 1 ) + C = ln [ e z ( 1 + e z ) ] + C =
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sviraju6d
Answered 2022-07-02 Author has 6 answers
The "right answer" is merely the simplification of the answer you found.
z + ln ( 1 + e z ) == ln ( 1 + e z ) e z + ln ( 1 + e z ) == e ln ( 1 + e z ) e z ( 1 + e z ) == 1 + e z e z + 1 == 1 + e z
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