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Evaluate the following integrals:

Question:

Evaluate the following integrals: $\int \mathrm{x}^{2} \sqrt{\mathrm{x}+2} \mathrm{dx}$

Solution:

Let $\mathrm{I}=\int \mathrm{x}^{2} \sqrt{\mathrm{x}+2} \mathrm{dx}$

Substituting, $x+2=t \Rightarrow d x=d t$,

$I=\int(t-2)^{2} \sqrt{t} d t$

$\Rightarrow I=\int\left(t^{2}-4 t+4\right) \sqrt{t} d t$

$\Rightarrow I=\int\left(t^{\frac{5}{2}}-4 t^{\frac{3}{2}}+4 t^{\frac{1}{2}}\right) d t$

$\Rightarrow I=\frac{2}{7} t^{\frac{7}{2}}-\frac{8}{5} t \frac{5}{2}+\frac{8}{2} t \frac{3}{2}+c$

$\Rightarrow I=\frac{2}{7}(x+2)^{\frac{7}{2}}-\frac{8}{5}(x+2)^{\frac{5}{2}}+\frac{8}{2}(x+2)^{\frac{3}{2}}+c$

Therefore, $\int x^{2} \sqrt{x+2} d x=\frac{2}{7}(x+2)^{\frac{7}{2}}-\frac{8}{5}(x+2)^{\frac{5}{2}}+\frac{8}{2}(x+2)^{\frac{3}{2}}+c$

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