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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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    @Thomas It looks to me like he is confused by the statement $F(x) = \int f(x)\ dx$, which is an indefinite integral with no bounds. The variable is just $x$, there's no integration to make it go away.2012-06-06

2 Answers 2

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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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    Got it. Thanks.2012-06-05
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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 $=$.2012-06-05