<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] The answer is both. Differentiate and you will get the original integrand in both cases.
Why? $\frac{1}{2}\ln{(\frac{5}{2} + e^x)} = \frac{1}{2}\ln{\frac{5+ 2e^x}{2}} = \frac{1}{2}(\ln{(5+ 2e^x)} - \ln{2})$, which differs from the other answer by a constant. Therefore, both antiderivatives are correct.
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