Notation for antiderivative

I came across the following:

$$F(x) = \int x^3 \cos(x)dx$$

where $F$ is understood to be a primitive of $x^3 \cos(x)$. I find this confusing, because of the "same" $x$ appearing on both sides of the equality. To me, $x$ is "integrated out" on the right side, and I prefer the notation:

$$F(x) = \int_{0}^{x} u^3 \cos(u) du$$

or possibly:

$$F = \int x^3 \cos(x) dx$$

without mentioning the variable for F.

Is the first notation widely used?

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it's plain wrong but often seen. –  user20266 Jun 5 '12 at 17:31
I don't think it's wrong. The indefinite integral notation is just poor because in a definite integral the variable is integrated out. –  Neal Jun 5 '12 at 17:35
@Neal Using $x$ on both sides of the equation, as variable on the lhs and bound to the integral on the rhs is the item the post was asking for, if I got it right. Indefinite integral I don't mind (a constant of integration would be a good thing (TM)). –  user20266 Jun 5 '12 at 17:37
Do you have problems with $(F(x))' = x^3 \cos(x)$. It's the same thing. Also, indefinite integral and a definite integral in interval [0, x] are somewhat different things. –  Karolis Juodelė Jun 5 '12 at 17:38
@KarolisJuodelė No, it's not the same thing. –  user20266 Jun 5 '12 at 17:40

The answer depends on what $\int f(x) \,dx$ means, about which there is no universal agreement. One interpretation is: the set of all functions whose derivative is $f$. If this definition is accepted, then $=$ should really be read as $\in$. This is the same convenient abuse of notation as in $\sqrt{x^2+1}=O(x)$. The other abuse is in writing $F(x)$ when you mean $F$, and this is also convenient at times.

So, this is how $F\in \int x^3\,dx$ becomes $F(x)=\int x^3\,dx$.

Notice that there is no integration involved in the above interpretation.

2nd interpretation: Someone may say that $\int f(x)\,dx$ is actually an integral, namely $\int_a^x f(t)\,dt$ with unspecified $a$. If you subscribe to this point of view, then $\sin x=\int \cos x\,dx$ is a true statement while $\sin x+5=\int \cos x\,dx$ is false.

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It's a bit scary when you have to wonder what $\int f(x) dx$ actually means :-) Better that it means only one thing that everybody agrees on, IMHO. –  Frank Jun 5 '12 at 17:52
@Frank which is what, exactly? –  user31373 Jun 5 '12 at 17:54
I believe your suggested first interpretation should be the only one accepted, or better, never use the shortcut, and be precise each time. I don't like maths to be open to "interpretation" = waste of time. –  Frank Jun 5 '12 at 17:56
@Frank I'm all for being precise, which is why I stated both definitions that people use, either consciously or not. When someone speaks of "variable of integration" etc, their words suggest there is integration involved in $\int\cos x\,dx$ (as in the 2nd interpretation). Personally, I stick with the first version: there is no integral whatsoever in $\int \cos x\,dx$. –  user31373 Jun 5 '12 at 18:09
Beyond the superficial notation issue, is the set of functions whose derivative is $f(x)$ different from the set ${\int_{a}^{x} f(t) dt, a \in \mathbb{R}}$ (supposing we are just talking about integrating on (subsets of) $\mathbb{R}$)? –  Frank Jun 5 '12 at 18:19

It is indeed widely used, as is the uglier notation $F(x) = \int^x f(w)dw$. $F(x) = \int f(x)dx$ means the family of primitives or antiderivatives: all $F$ such that $F'(x)=f(x)$. Nothing to worry about.

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It is readily understandable, IMHO, although I see it as a shortcut with some confusion on $x$ on both sides of $=$. –  Frank Jun 5 '12 at 17:47