For the integral of x (4-x)^(1/2) dx is more conducive to integrate by de-radicalization, letting the radicand be u^2. So, x = u^2 +4; dx = 2u du and the radical expression becomes u. It bangs out pretty simply.
Sir Eddie there is a rule created by me called as I LATE ... To choose U and V in integration product rule I = Inverse function L= LOG A= algebraic T = trigonometric E = exponential Which even come first is taken as U other become V
Because the notation of sin^(-1)(x) has nothing to do with 1/sin(x). It's a confusing notation, because it refers to the function inverse, instead of the reciprocal. 1/sin(x) is called cosecant, and even it can't just be integrated with the method you propose. The chain rule works for differentiating any function, but for integrating, it isn't as straight forward. You have to first spot the derivative outside the function, in order to use the chain rule in reverse. A derivative that is a simple constant can always be produced by multiplying by 1 in a fancy way, but any other derivative, not so much.
