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Let $f,g : \mathbb{R} \to \mathbb{R}$. Is it acceptable to write $\int_a^b fg$ instead of $\int_a^b f(x)g(x)\,dx$ (i.e., would it throw others off while reading it)? The (lack of an) indeterminate is unambiguous since the function $fg$ only accepts one argument, but I've never seen others write it like that.

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I have seen it all too often, from students who then can't handle substitution. – André Nicolas Nov 27 '12 at 6:49
@AndréNicolas: Can you explain? I don't see the problem with substitution: $\int_a^b (f' \circ g) g' = \int_{g(a)}^{g(b)} f$. – Snowball Nov 27 '12 at 7:15
I believe you can handle it. Was observing from grading too many calculus final exams that there is significant correlation between leaving out $dx$ and getting the wrong answer for $\int \frac{dx}{1+3x}$. – André Nicolas Nov 27 '12 at 7:20
up vote 1 down vote accepted

I wouldn't do that, a) because it looks ugly to me, and, perhaps more importantly, b) because it prevents you from writing things like

$$ \int_0^{\pi}fg\sin\theta\,\mathrm d\theta=\int_{-1}^1fg\,\mathrm d\cos\theta\;. $$

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