∫2x(x+2)^(1/2) dx evaluated from 0->2

First make a substitution so we can apply the power rule ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C more simply. Can see u=x+2 means (x+2)^1/2 -> u^1/2 and so will help us apply this rule. Change dx/du=1 implies dx = du, and x=0 => u=2, x=2 => u =4 so we now integrate from 2 -> 4.
x=u-2 gives ∫2x(x+2)^1/2 dx = ∫2(u-2)(u)^1/2 dx = 2∫u^3/2 du - 4 ∫u^1/2 du. We can now apply the power law above with the new integration limits to obtain the answer (32)/(15) ( 2 + (2)^(1/2) ).