Ho To Perform U-Substitution On Given Intergal

Question

$$\int{x^2\sqrt{2+x}}\ {dx}$$

I haven't been able to find what u should be in this integral, where should I start?

I've gotten as far as:

let $u = 2 + x$; $du=\frac{1}{x}dx$

Answer 1

Let $u=2+x\;\Rightarrow\; x=u-2\;\Rightarrow\; dx=du$, then $$ \begin{align} \int x^2\sqrt{2+x}\ dx&=\int (u-2)^2\sqrt{u}\ du\\ &=\int (u^2-4u+4)\ u^{\frac12}\ du\\ &=\int \left(u^{\frac52}-4u^{\frac32}+4u^{\frac12}\right)\ du \end{align} $$ or let $u^2=2+x\;\Rightarrow\; x=u^2-2\;\Rightarrow\; dx=2u\ du$, then $$ \begin{align} \int x^2\sqrt{2+x}\ dx&=\int (u^2-2)^2u\cdot 2u\ du\\ &=2\int (u^4-4u^2+4)\ u^{2}\ du\\ &=2\int \left(u^{6}-4u^{4}+4u^{2}\right)\ du. \end{align} $$