integrate function with change of variable Find the primitive of $\;\displaystyle \int x^2 \sqrt{x+1}$ $dx$
So (...)
$u = x + 1 \quad \iff \quad u - 1 = x$
$u' = 1 \quad \iff \quad \frac{du}{dx} = 1 \rightarrow \;du = dx$
$$\int(u - 1)^2 . u^\frac12 \; du \;\;= \;\; {{(u-1)^3}\over3} \cdot {u^{3/2}\over{3/2}} + C \;\; =\;\; {1\over3} x^3\cdot{2\over3}(x+1)^{3/2} + C$$ 
Is this correct?

You have it right up to $$\int(u-1)^2 u^{1/2} du$$. However it is not in general correct that $\int (f\cdot g)(x)dx=\int f(x)dx \cdot \int g(x) dx $, that is, integration does not "distribute" over multiplication.
