<s>[INST] Weird integration issue: $\ln(x+1)=\ln(2x+2)$ ?! Weird integration issue: Using $(\ln[f(x)])'=\frac {f'(x)}{f(x)}$ we get that $\int \frac{2\,dx}{2x+2}=\ln(2x+2)$. Yet, $\int \frac{2\,dx}{2x+2}= \int\frac{dx}{x+1}=\ln(x+1)$ using the same rule as earlier.
What is wrong here?
 [/INST] Remember that:
$$\int \frac{2}{2x+2}dx \neq \ln(2x+2)$$
But:
$$\int \frac{2}{2x+2}dx = \ln(2x+2)+C$$
where $C$ is constant, so:
$$\int \frac{2}{2x+2}dx = \ln(2x+2)+C=\ln(2(x+1))+C=\ln(x+1)+\ln 2+C=\ln(x+1)+(\ln 2+C)=\ln(x+1)+C_1$$
So both calculations are almost correct, but you forgot about constant.
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