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 hard-and-fast 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, over-thinking 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.