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

$\int\frac{(1+\sqrt{\text{x}})^2}{\sqrt{\text{x}}}\text{dx}$
$\text{Let},1+\sqrt{\text{x}}=\text{t}$
$\Rightarrow\frac{1}{2\sqrt{\text{x}}}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\frac{\text{dx}}{\sqrt{\text{x}}}=2\text{dt}$
$\text{Now},\int\frac{(1+\sqrt{\text{x}})^2}{\sqrt{\text{x}}}\text{dx}$
$=2\int\text{t}^2\text{dt}$
$=\frac{2}{3}\text{t}^3+\text{C}$
$=\frac{2}{3}(1+\sqrt{\text{x}})^3+\text{C}$

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