^2 x+2 dx - Vidyadip

Question

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

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Answer

$\int\text{x}^2\sqrt{\text{x}+2}\text{ dx}$
Let $\text{x}+2=\text{t}$
$\Rightarrow\text{x}=\text{t}-2$
$\Rightarrow\text{dx}=\text{dt}$
Now, $\int\text{x}^2\sqrt{\text{x}+2}\text{ dx}$
$=\int(\text{t}-2)^2\sqrt{\text{t}}\text{ dt}$
$=\int(4^2-4\text{t}+4)\text{t}^\frac{1}{2}\text{dt}$
$=\int\Big(\text{t}^{2+\frac{1}{2}}-4\text{t}^{1+\frac{1}{2}}+4\text{t}^{\frac{1}{2}}\Big)\text{dt}$
$=\int\Big(\text{t}^{\frac{5}{2}}-4\text{t}^{\frac{3}{2}}+4\text{t}^{\frac{1}{2}}\Big)\text{dt}$
$=\Bigg[\frac{\text{t}^{\frac{5}{2}+1}}{\frac{5}{2}+1}\Bigg]-4\Bigg[\frac{\text{t}^{\frac{3}{2}+1}}{\frac{3}{2}+1}\Bigg]+4\Bigg[\frac{\text{t}^{\frac{1}{2}+1}}{\frac{1}{2}+1}\Bigg]+\text{c}$
$=\frac{2}{7}\text{t}^{\frac{7}{2}}-\frac{8}{5}\text{t}^{\frac{5}{2}}+\frac{8}{3}\text{t}^{\frac{3}{2}}+\text{C}$
$=\frac{2}{7}(\text{x}+2)^\frac{7}{3}-\frac{8}{5}(\text{x}+2)^\frac{5}{2}+\frac{8}{3}(\text{x}+2)^\frac{3}{2}+\text{C}$

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