Integrate y^2 cos3y
I though of using product rule
1/3sin3y y^2 - integral of 1/3sin3y 2y
it looks quite complicated already.
Use integration by parts - twice. (Assuming you have correctly used it once, the remaining integral is obviously simpler. One more integration by parts and the remaining integral will be trivial).
And, yes, what you have is simpler, not more complicated. Before you had $\displaystyle y^2$ multiplying the trig function, now only "y". One more integration by parts should remove that "y" entirely.