1. INT(Sqrt(3 - 2x)x^2) dx
This can be solved without integration by parts. You might find it hard to believe, but we can actually solve this by u substitution:
Let u = 3 - 2x ... Why would I do that? You'll see .
du = -2dx --> dx = -du/2
Before I go on, let's solve the u = 3 - 2x for x (we'll need this in a second): x = 1/2(3 - u)
Ok, now watch my slight of hand as I do some math-magic on this integration...
INT(Sqrt(u)x^2) -du/2 ... Now you see an x^2,
-1/2 INT(Sqrt(u)[1/2(3 - u)]^2) du ... Now you don't. ... Guess where it went. (Hint: x = 1/2(3 - u))
From here, I think you can solve it the rest of the way, just multiply out the [1/2(3 - u)]^2 then multiply this to the Sqrt(u).