Methods for choosing $u$ and $dv$ when integrating by parts? When doing integration by parts, how do you know which part should be $u$ ? 
For example, 
For the following:
$$\int x^2e^xdx$$
$u = x^2$?
However for:
$$\int \sqrt{x}\ln xdx$$
$u = \ln x$?
Is there a rule for which part should be $u$ ? As this is confusing.

My general principle is "Which bit gets nicer to work with when differentiated than when integrated?", which roughly lines up with the LIATE approach:

*

*The derivative of $\ln x$ is $\frac{1}{x}$, which will interact nicely with polynomial terms, whereas the integral is some weird rubbish on the order of $x \ln x$ which looks terrible.

*The derivatives of the inverse trigonometric functions tend to be rational functions or square roots of rational functions, which again tend to interact with polynomial terms decently, whereas the integrals are horrible things with logarithms in them.

*Algebraic stuff (polynomials) differentiates into other algebraic stuff, and if you're lucky it will get simpler as it goes (but not always).

*Trig and exponential functions tend to sit at about the same complexity no matter how much you differentiate or integrate them. So if you're left picking them as your $u$ it probably means you're going to be setting up a recurrence relation of some kind.

