When Integrating by parts, how do you know which part to make "u" and "dv/dx"?

you need to think about how the integral will be simpliefied, basically you are trying to get to a state where only one function of x resides in the integral. so if you have an x² sin3x dx inside the integral, you cannot easily get rid of the sin3x. the thing you need to do is be rid of the x². to do this, we need to differentiate twice to get to a constant, 2. therefore first, we would set x² as u, and sin3x as dv/dx. after 2 runs through integration by parts, the function inside the integral will be easily found. Here we end up with du/dx = 2x and v = 1/3 cos3x. We then get 1/3 x²cos3x - integral of 2/3 xcos3x dx. using integration by parts again we get du/dx = 2/x, meaning now inside the integral we have a single function only consisting of a multiplier and a sin3x term.