Before I start to go over what I've been doing in regards to trigonometric substitution, it'd be helpful to explain integration by parts.
Integrating by parts is based off the product rule and can be summed up like this:
where int = an integral.
int( u dv) = uv - int( v du).
Essentially you break down the initial integral into the parts u and dv with dv being the most complicated function that you are able to easily integrate.
It's best to set it up like so:
u = ???? dv = ????
du = ???? v = ????
If you maintain this order, it's easy to integrate regardless of what letters you are using for u and v.