I'm confused if it is possible to evaluate this class of integral if it is a correct formula :
QUESTION:
What's this : $\displaystyle\int{\frac{1}{{x}dx}}$ equal if it is a correct formula ?
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zeraoulia rafik
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2Do you really mean $\int \frac 1{x,\mathrm{d}x}$, or rather $\int \frac 1{x}\mathrm{d}x$? – DominikS Jul 25 '17 at 23:21
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yes, i seek if this suitable formula with measure theory and if it is how i can evaluate it – zeraoulia rafik Jul 25 '17 at 23:21
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8The notation $\int \frac{1}{x,dx}$ is nonsense. – Mark Viola Jul 25 '17 at 23:23
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5$\frac 1{\mathrm{d} x}$ does not correspond to a measure, e.g. additivity is violated. – DominikS Jul 25 '17 at 23:24
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2@MarkViola Actually, it is not. Unusual, but it can be made make perfect sense including being a particular case of a very common notation. – Nina Simone Jul 25 '17 at 23:24
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In that case it would be helpful to know where you encountered this integral. Even if it could be interpreted in a meaningful way, it might still be just a typo. – Andre Jul 25 '17 at 23:27
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5It is meaningless. See: https://math.stackexchange.com/questions/304549/how-to-evaluate-a-definite-integral-that-involves-dx2/304557#304557 – Thomas Andrews Jul 25 '17 at 23:28
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6@NinaSimone "Unusual, but it can be made make perfect sense including being a particular case of a very common notation" I am surprised. Would you care to provide some more explanations? – Did Jul 25 '17 at 23:33
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4@NinaSimone Oh? Please educate us on this "unusual" notation. – Mark Viola Jul 25 '17 at 23:37
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It is not correct notation. – David G. Stork Jul 26 '17 at 00:51
3 Answers
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I would be surprised if it was a correct formula. If there was, it would at least not be the standard way to write integrals. In the notation of an integral, the $dx$ part is reminiscent of the fact that the integral stems from a summation $$ \int f(x) dx \approx \sum_{i} f(x_{i}) \Delta x_{i} $$ where $\Delta x_{i} \rightarrow 0$. So I would say - without knowing further context - that the formula is a typo and that the person who wrote it did unintentionally put the $dx$ under the fraction line.
What is probably meant is $$ \int \frac{1}{x} dx, $$ which is a standard integral with stem function $\ln(x)$.
Hope that helps!
Andre
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In order for the notation $\int f(x)\frac 1{\mathrm{d} x}$ to make sense, the map $A\mapsto \int_A\frac 1{\mathrm{d} x}$ would have to be a measure on $\mathbb R$ (where $A\subset \mathbb R$ measurable).
But an expression of the form "1 over Lebesgue measure" will not be a measure any more, it violates additivity, for example.
DominikS
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Almost surely, there is a serious typo in the expression that motivates the question. That is, an expression $\int_X {f(x)\over g(x)\,dx}$ does not have any clear interpretation in contemporary mathematics, ... other than as a typo for $\int_X {f(x)\over g(x)}\;dx$. After all, the $\int_X...dx$ is really just a historical artifact for a linear functional, perhaps extended by continuity from $C^o_c(X)$ to a larger class of functions.
paul garrett
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