Use the substitution \(u=\sqrt{x+2}\) to find \(\int\frac{x}{\sqrt{x+2}}\,\mathrm{d}x\).[4]
\(u=\sqrt{x+2}\Rightarrow x=u^2-2,\quad \mathrm{d}x=2u\,\mathrm{d}u\)
\(\int\frac{x}{\sqrt{x+2}}\,\mathrm{d}x=\int 2(u^2-2)\,\mathrm{d}u\)
\(=\frac{2}{3}u^3-4u+C\)
\(=\frac{2}{3}(x+2)^{3/2}-4\sqrt{x+2}+C\)
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