<s>[INST] Integral of $\frac{e^x}{5+2e^x}$ Regarding the integral of this term$\frac{e^x}{5+2e^x}=\frac{e^x}{2(\frac{5}{2}+e^x)}$
Is the answer $\frac{1}{2} \ln(\frac{5}{2} +e^x)$ or $\frac{1}{2} \ln(5+2e^x)$?
When I substitute $u= \frac{5}{2}+e^x$ I get the first one, when it's $u=5+2e^x$, then it is the second one...
 [/INST] Since
$$\frac{\mathsf d}{\mathsf dx}\left[\frac12\log(5+2e^x)\right] = \frac{2e^x}{2(5+2e^x)}=\frac{e^x}{5+2e^x}, $$
it follows that $\frac12\log(5+2e^x)$ is an antiderivative of $\frac{e^x}{5+2e^x}.$
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