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.

Forget cute (or weird) acronyms or tables. You are focusing on the wrong thing: differentiation is easy, while integration is hard, so you should ask not what to pick as $u$, but what to pick as $dv$. The answer is: choose as $dv$ the most complicated expression in the integrand that you currently know how to integrate. 
For example, you asked about integrating $x^2e^x$. Between $x^2$ and $e^x$ the factor $e^x$ is more sophisticated and you can integrate it, so let $dv = e^x dx$ and then $u = x^2$. You also asked about integrating $\sqrt{x}\ln x$. For students the antiderivative of $\sqrt{x}$ is known but the antiderivative of $\ln x$ is not, so let $dv = \sqrt{x} dx$ and then $u = \ln x$.
When this tip of how to pick $dv$ rather than $u$ was passed on to me, I never had a problem applying integration by parts afterwards. This method requires no memorization of rules, but just experience integrating to recognize one factor as being more complicated than another.
