Best way to organize multiple integrations by parts? It is always hard for me to write the integration by parts on paper when it requires a double or triple,... , etc. integration by parts. What is your way to write it down best that doesn't look confusing?
 A: Have you heard of tabular integration? This is a nice and neat way to organize your integration by parts:
\begin{array}{c|c|c}
\color{red}{u} & dv & \text{sign} \\\hline
\color{blue}{u'} & \color{red}{v} & \color{red}{+} \\\hline
\color{green}{u''} & \color{blue}{w=\int v} & \color{blue}{-} \\\hline
\dots & \color{green}{\int w} & \color{green}{+}
\end{array}
and the pattern continues until you get $0$ in the left column. You take successive derivatives on the left column and successive integrals in the middle column; the signs alternate on the right starting with $+$. The way you put this into practice is that you match all the colors, so you end up with
$$
\int u\,dv = \color{red}{+\, uv} \color{blue}{- u'\int v\,dx} \color{green}{+\dotsb}
$$
Here is an example where you would have to perform integration by parts multiple times simplified with tabular integration:
$$
\int x^3e^x\,dx=\, ?
$$
Let $u = x^3$ and $dv = e^x\,dx$:
\begin{array}{c|c|c}
\color{red}{x^3} & e^x & \text{sign} \\\hline
\color{blue}{3x^2} & \color{red}{e^x} & \color{red}{+} \\\hline
\color{green}{6x} & \color{blue}{e^x} & \color{blue}{-} \\\hline
\color{brown}{6} & \color{green}{e^x} & \color{green}{+} \\\hline
0 & \color{brown}{e^x} & \color{brown}{-}
\end{array}
Hence,
$$
\int x^3e^x\,dx= \color{red}{x^3e^x}\color{blue}{-3x^2e^x}\color{green}{+6xe^x}\color{brown}{-6e^x} + C,
$$
where $C$ is an additive constant. You should check that you get the same result either way. This method is also demonstrated in the film Stand and Deliver about Los Angeles high school calculus teacher Jaime Escalante.
A: The formula for partial integration is given by:
$$\int fdg = fg - \int gdf$$
So, I always write integrals in this form when performing partial integration. Since an example can say more than a thousand words:
$$\int xe^xdx = \int xd(e^x) = xe^x - \int e^x dx = xe^x - e^x + c = e^x(x -1) + c$$
A: Make sure to write it out with the definition.  (Literally write it down in terms of u and v, without even using your problem.  Then start mechanically working it through showing step after step.)  I think you are trying to skip some steps and some writing and then that bites you in the butt.
Remember how you would move stuff around in pre-algebra to solve a simple equation for x?  Adding numbers and subtracting variables from each side and dividing by the divisor.  Step by step by step.  That was needed when the thing was new to you.  Sure you got to a point later where you didn't need to be so mechanical.  But that came from a lot of practice and exposure and doing it the mechanical way first.  
Same for integration by parts.  Do it with all the step by step writing.  When you have done it enough, you will just want to not write so much but you will know the process cold, so no big deal.  But at this point...you don't know it cold.
