TITLE: U-substitution for integral of 1/(1+e^x)dx. What am I doing wrong?
QUESTION [9 upvotes]: Here is my work, with the right answer. I feel like every step is right, but somehow I am getting the wrong answer. How?
$$ \int \frac{1}{1+e^z}dz = \int\frac{1}{e^z(\frac{1}{e^z} + 1)}dz  
=\int\frac{1}{\frac{1}{e^z} + 1}\frac{1}{e^z}dz $$
subbing $u = \frac{1}{e^z} + 1$, $\frac{du}{dz} = -e^{-z}dz \implies -du = \frac{1}{e^z}dz$ so
$$\int \frac{1}{1+e^z}dz = -\int\frac{1}{u}du = -\ln(u) + C = -\ln(e^{-z}+1) + C$$
But the right answer is $z - \ln(1+e^z)+C$

REPLY [4 votes]: The "right answer" is merely the simplification of the answer you found.
$$
z+\text{ln}\left(1+e^{-z}\right)==\text{ln}\left(1+e^z\right)\\
e^{z+\text{ln}\left(1+e^{-z}\right)}==e^{\text{ln}\left(1+e^z\right)}\\
e^z\left(1+e^{-z}\right)==1+e^z\\
e^z+1==1+e^z
$$