The integral of $\frac {1}{x}$ and basic differential equations In introductory calculus, when solving basic separable ordinary differential equations, we often use the following "fact" 
$$\int \frac 1x \ dx = \ln|x| + C$$
This "fact", however, is slightly misleading. The actual integral is the following: 
$$\int \frac 1x \ dx = \begin{cases}\ln \left|x \right| + C_1 & x < 0\\
\ln \left|x \right| + C_2 & x > 0
\end{cases}$$
My question, is, simply, as far as basic (that is, AP Calculus level) ordinary differential equations go, will using the former integral ever yield imprecise solutions? 
EDIT: I have received three close votes on this question since it is "opinion based". I do not understand how---I have a concrete mathematical question, the answer to which is certainly not an "opinion". I am presuming this is resulting from my usage of the word "false" in describing the first equation, which I have changed to "slightly misleading". If there are any other problems with this question, I would certainly love to know in the comments. 
 A: Your question, as is, will probably have artificial answers as result. But it is "equivalent" to a potentially more interesting and general problem: 

Is it possible for non-artificial problems to arise whenever we use the fact that a function has derivative identically zero to arrive at the fact that it is constant without verifying that the set is connected?

And this can indeed happen in a non-artificial situation. We have as an example an answer which I gave here in MSE. I started working on a subset of $M_n(\mathbb{R})$, arrived at the fact that the function had zero derivative, hence it should be constant, but didn't verify that the set was connected, which was not at all justified (and even made this question exist). However, using $M_n(\mathbb{C})$ I was able to arrive at the correct result (and in particular it holds for $M_n(\mathbb{R})$.
Of course, one could argue that the answer itself to that question is convoluted and artificial, but this technique of using the fact that a function with derivative identically zero is constant on a connected set is ubiquitous; the answer linked being only the first example that came to mind.
