Am I allowed to add and subtract expressions whose domain might not be $ \mathbb{R} $

Question

We all know the trick of adding and subtracting an expression, for example to calculate:

$ \int \frac{2x}{2x + 1} \, dx = \int \frac{2x + 1 - 1}{2x + 1} \, dx = \int \frac{2x + 1}{2x + 1} \, dx + \int \frac{1}{2x + 1} \, dx = \int 1 \, dx + \int \frac{1}{2x + 1} = x + \frac{1}{2} ln\left|2x + 1\right| + C $.

But, is it OK to add expressions such as $ \frac{1}{x} $ or $ ln(x) $, whose domain is not $ \mathbb{R} $.

MathJax hint: if you put a backslash before common functions you get the right font and spacing, so \ln x gives $\ln x$ instead of ln x giving $ln x$ — Ross Millikan, Jul 01 '22 at 15:06

score 1 · Answer 1 · answered Jul 01 '22 at 15:05

1

Yes, but the validity of your solution is restricted to the range of $x$ that all your manipulations support. In your example, $x$ is restricted by the problem to $\Bbb R \setminus \{-\frac 12\}$ by the denominator of the fraction and all the steps are valid over that domain. Had you added and subtracted $\frac 1x$ somewhere your solution would not be valid at $0$. Depending on the problem, you might not care. There might be some reason you already know $x \gt 0$. Otherwise, you might solve the problem separately for $x=0$ avoiding that step and get one answer valid at $0$ and another valid elsewhere.

answered Jul 01 '22 at 15:05

Ross Millikan

374,822

How can one evaluate an integral “_at x=0_”? – insipidintegrator Jul 01 '22 at 16:46
@insipidintegrator: As I read it, the question is much more general than integrals. If the adding and subtracting $\frac 1x$ were inside an integral you would have to deal with the possible singularity in the integrand. – Ross Millikan Jul 01 '22 at 18:20
Oh…ok….. gently goes out the back door, and makes a run for the hills where people discuss easy integrals – insipidintegrator Jul 01 '22 at 18:22

Am I allowed to add and subtract expressions whose domain might not be $ \mathbb{R} $

1 Answers1