Mathematics

Evaluate the given integral.
$$\displaystyle\int{\dfrac{{\left(1+x\right)}^{2}}{\sqrt{x}}dx}$$


SOLUTION
$$I=\displaystyle\int{\dfrac{{\left(1+x\right)}^{2}}{\sqrt{x}}dx}$$

$$=\displaystyle\int{\dfrac{\left(1+{x}^{2}+2x\right)}{\sqrt{x}}dx}$$

$$=\displaystyle\int{\left({x}^{-\frac{1}{2}}+{x}^{2-\frac{1}{2}}+2{x}^{1-\frac{1}{2}}\right)dx}$$

$$=\displaystyle\int{\left({x}^{-\frac{1}{2}}+{x}^{\frac{3}{2}}+2{x}^{\frac{1}{2}}\right)dx}$$

$$=\displaystyle\int{{x}^{-\frac{1}{2}}dx}+\displaystyle\int{{x}^{\frac{3}{2}}dx}+2\displaystyle\int{{x}^{\frac{1}{2}}dx}$$

$$=\dfrac{{x}^{\frac{-1}{2}+1}}{\dfrac{-1}{2}+1}+\dfrac{{x}^{\frac{3}{2}+1}}{\dfrac{3}{2}+1}+2\times\dfrac{{x}^{\frac{1}{2}+1}}{\dfrac{1}{2}+1}+c$$

$$=\dfrac{{x}^{\frac{1}{2}}}{\dfrac{1}{2}}+\dfrac{{x}^{\frac{5}{2}}}{\dfrac{5}{2}}+2\times\dfrac{{x}^{\frac{3}{2}}}{\dfrac{3}{2}}+c$$

$$=2\sqrt{x}+\dfrac{2}{5}{x}^{\frac{5}{2}}+\dfrac{4}{3}{x}^{\frac{3}{2}}+c$$
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