Integration by Parts
After $u$substitution, integration by parts is the most
important technique to learn. It converts a hard integral
$\int u\,dv$ into an easier integral $ uv  \int v\,du$. The
tricky thing is figuring out what to pick for $u$ and $dv$.
We want to choose $u$ and $dv$ so that $v\,du$ is easier to
integrate than $u\,dv$. While there aren't really
hardandfast rules, the following hints help.
Integration by parts: Some hints
for choosing $u$ and $dv$
 Some functions do not have an antiderivative that we
know. These must be $u$.
 Some functions get a lot simpler when differentiated
or a lot more complicated when integrated. These
almost always should be $u$.
 Some functions get a little simpler when
differentiated, and some functions, like $e^x$ and
$\sin(x)$, don't get simpler at all. Whether
these are $u$ or $dv$ will depend on the rest of the
integrand.
 Some functions actually get simpler when integrated
and are often part of $dv$.

Remember, you can try various $u$ and $dv$, then differentiate
and antidifferentiate. If your second integral looks worse,
try again. Sometimes, we may have to integrate by parts
twice to get our answer.
Example: Compute
$\displaystyle\int\frac{\ln(x)}{x^3}\,dx. $
Solution: We suspect this is a good integration by parts
problem since substitution won't work, and since the integrand can
be written as the product of two functions, $\ln n$ and
$\frac{1}{x^3}$. Since we don't know the antiderivative of
$\ln x$, we take $u=\ln(x)$. We get $$
\int\frac{\ln(x)}{x^3}\,dx \overset{\fbox{$\,u\,=\,\ln (x)\quad
v\,=\,\frac{1}{2x^2}\\du\,=\,\frac{1}{x}\,dx\,\,\,
dv\,=\,\frac{1}{x^3}\,dx$}\\}{=}\frac{\ln(x)}{2x^2}\left(\int
\frac{1}{2x^3}\,dx\right)=\frac{\ln(x)}{2x^2}\frac{1}{4x^2}+C.
$$
The two most important things to
remember about integration by parts are
1) when to use this technique, In general, an
integrand that is the product of two functions is a good candidate
for parts. If you do not see a substitution, and it is not
an obvious trigonometric integral, then parts is good to
try.
and
2) do not be afraid of trial and
error. Once you decide to try parts,
overthinking what to pick for $u$ and $dv$ takes longer than just
trying something. If your trial doesn't work, mark through
it and try something else. (Don't erase, or you'll forget
what you've tried!) The more you practice, the better feel
you'll get for how to choose.
