Is there a short symbol that denotes integration? I want to illustrate partial integration, see below. With derivatives we can just write $(term)'$. Is there something similar for integration? The best I could come up with is $\int(term)\mathrm{d}x$. But I find it not as readable as $()'$: first the $\int$ has a huge vertical expansion obstructing the reading flow, second I need to enclose it with even more characters denoting the variable by which we integrate.
$$
\begin{align*}
\int_2^{10} x\cdot\ln(x) \,\mathrm{d}x &= \left[ \int(x)\mathrm{d}x \cdot\ln(x)\right]_2^{10} - \int_2^{10} \int(x)\mathrm{d}x \cdot \ln(x)' \,\mathrm{d}x \\
&= \left[ \frac12 {x^2}\cdot\ln(x)\right]_2^{10} - \int_2^{10} \frac12 {x^2} \cdot\frac1{x} \,\mathrm{d}x \\
\end{align*}
$$
Is there something better?
 A: If the goal is to be readable and understandable by others: use the standard notations. So that people don't have to put too much effort in order to understand your equations. Attention $$\text{compact notation $\neq $ easy to read}$$
Sometimes it helps, sometimes it just makes things worst even more if the reader is not familiar with the notation used in the compact version. => use standard notations.
For the integration by part, just turn it in the other way:
$$ \int_a^b u'(x) v(x) dx = \Big[u(x)v(x)\Big]_{x=a}^{x=b}-\int_a^bv'(x)u(x)dx$$
(no need for a special sign for the indefinite integral here). You could even write:
$$ u'(x)=x \qquad v(x)=\ln(x) \qquad \implies \qquad u(x)=\frac{x^2}{2}+c \qquad v'(x)=\frac{1}{x}$$
and proceed your integration. It's a little bit longer to write but much more clear and it shows that you understand what you're doing. In particular, in an exam, if you made a mistake when computing $u$ or $v'$ then the corrector sees at least that you understood what is integration by parts and how it works (so that he can give you a few points if he is nice).
