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?

The substitutions will work fine...but your evaluation is problematic. 
Most problematic is the fact that you are integrating each factor of a product in the integrand, and expressing this as the product of integrated factors: which you cannot do, unless you are using, say, integration by parts, which proceeds much differently: . I.e. $$\int [f(x)\cdot g(x)]\,dx \;\neq \;\int f(x) \,dx \cdot \int g(x)\,dx$$

So, let's start from the point after which we've substituted:
$$\int \underbrace{(u - 1)^2}_{\text{expand}} . u^{1/2} \; du \; = \;\int \underbrace{(u^2 - 2u + 1)u^{1/2}}_{\text{distribute}} \,\;du = \int \left(u^{5/2} - 2u^{3/2} + u^{1/2}\right) \,du$$
Now integrate, and then back-substitute.
