0
$\begingroup$

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.

  • 2
    I have seen it all too often, from students who then can't handle substitution.2012-11-27
  • 0
    @AndréNicolas: Can you explain? I don't see the problem with substitution: $\int_a^b (f' \circ g) g' = \int_{g(a)}^{g(b)} f$.2012-11-27
  • 1
    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 $\int \frac{dx}{1+3x}$.2012-11-27

1 Answers 1

1

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

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