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

Question

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

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Answer

$\int\frac{(\text{x}+1)(\text{x}-2)}{\sqrt{\text{x}}}\text{dx}$
$=\int\bigg(\frac{\text{x}^2-2\text{x}+\text{x}-2}{\sqrt{\text{x}}}\bigg)\text{dx}$
$=\bigg(\frac{\text{x}^2-\text{x}-2}{\sqrt{\text{x}}}\bigg)\text{dx}$
$=\int\Big(\text{x}^\frac{3}{2}-\text{x}^\frac{1}{2}-2\text{x}^\frac{-1}{2}\Big)\text{dx}$
$=\Bigg[\frac{\text{x}^{\frac{3}{2}+1}}{\frac{3}{2}+1}\Bigg]-\Bigg[\frac{\text{x}^{\frac{1}{2}+1}}{\frac{1}{2}+1}\Bigg]-2\Bigg[\frac{\text{x}^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}\Bigg]+\text{C}$
$=\frac{2}{5}\text{x}^\frac{5}{2}-\frac{2}{3}\text{x}^\frac{3}{2}-4\text{x}^\frac{1}{2}+\text{C}$

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