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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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    it's plain wrong but often seen.2012-06-05
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    I don't think it's wrong. The indefinite integral notation is just poor because in a definite integral the variable is integrated out.2012-06-05
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    @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)).2012-06-05
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    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.2012-06-05
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    @KarolisJuodelė No, it's not the same thing.2012-06-05
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    @Neal - yes, using $x$ on both sides is to me very confusing, although I can understand the shortcut. I like the equality to mean equality.2012-06-05
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    @Karolis - agree with Thomas, it's not the same. I believe the correct definition of the antiderivative is precisely the second form above $F(x) = \int_{0}^{x} x^3 cos(x) dx$, which is not apparent in the first notation.2012-06-05
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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

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