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I'm just wondering if $\int\,\mathrm dx$ can be written instead of $\int1\,\mathrm dx$.

This sometimes comes up while finding the integrating factor of a suitable differential equation, when the coefficient of $y$ is $1$ and the integrating factor is thus $e^{\int1\,\mathrm dx}=e^x$.

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    $\begingroup$ Yes, usually the $1$ is omitted, just like when we write $x$ instead of $1x$. $\endgroup$ – user2825632 Sep 16 '16 at 21:33
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    $\begingroup$ @user2825632 Thank you very much. Please post this as an answer. $\endgroup$ – Skeleton Bow Sep 16 '16 at 21:34
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Yes, usually the $1$ is omitted, just like when we write $x$ instead of $1x$.

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