Evaluate the following integrals: (1+ x )^2 x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{(1+\sqrt{\text{x}})^2}{\sqrt{\text{x}}}\text{dx}$

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Answer

$\int\frac{(1+\sqrt{\text{x}})^2}{\sqrt{\text{x}}}\text{dx}$
$=\int\frac{1+\text{x}+2\sqrt{\text{x}}}{\text{x}^{\frac{1}{2}}}\text{dx}$
$\int\text{x}^{\frac{-1}{2}}+\int\text{x}^{\frac{1}{2}}\text{dx}+2\int\text{dx}$
$=2\sqrt{\text{x}}+\frac{2}{3}\text{x}^{\frac{3}{2}}+2\text{x}+\text{C}$
$\therefore\ \int\frac{(1+\sqrt{\text{x}})^2}{\sqrt{\text{x}}}\text{dx}=2\sqrt{\text{x}}+\frac{2}{3}\text{x}^{\frac{3}{2}}+2\text{x}+\text{C}$

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