How do I integrate $\int\dfrac{e^x - e^{-x}}{e^x + e^{-x}}\,dx$?

You have $du$ error. It is $du=(e^x-e^{-x})dx$ so after variable change integrating just $du/u$ which is $\ln(u).$

Your $du$ is wrong. $du=e^x-e^{-x}dx$

Your integral just becomes $\int \frac1u du$. This becomes $\ln(e^x+e^{-x})+C$. You can use algebra a bit here to get your result.

You made a small mistake, $du$ is actually equal to $e^x-e^{-x}$, so you have $\int \dfrac {du}{u}$