Evaluate the following integrals: ^2_0x 2-x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^2_0\text{x}\sqrt{2-\text{x}}\text{ dx}$

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Answer

Let $\text{I}=\int\limits^2_0\text{x}\sqrt{2-\text{x}}\text{ dx}$
$=\int\limits^2_0(2-\text{x})\sqrt{2-2+\text{x}}\text{ dx}$
$=\int\limits^2_0(2-\text{x})\sqrt{\text{x}}\text{ dx}$
$=\int\limits^2_0\big(2\sqrt{\text{x}}-\text{x}\sqrt{\text{x}}\big)\text{dx}$
$=\int\limits^2_0\Big(2\text{x}^{\frac{1}{2}}-\text{x}^{\frac{3}{2}}\Big)\text{dx}$
$=\Bigg[2\frac{\text{x}^{\frac{3}{2}}}{\frac{3}{2}}-\frac{\text{x}^{\frac{5}{2}}}{\frac{5}{2}}\Bigg]$
$=\bigg[\frac{4}{3}\text{x}^{\frac{3}{2}}-\frac{2}{5}\text{x}^{\frac{5}{2}}\bigg]^2_0$
$=\frac{8\sqrt{2}}{3}-\frac{8\sqrt{2}}{5}$
$=\frac{16\sqrt{2}}{15}$

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