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.
