Evaluate the following definite integrals: _ 4 ^91 x dx - Vidyadip

Question

Evaluate the following definite integrals:
$\int_{4}^\limits{9}\frac{1}{\sqrt{\text{x}}}\text{ dx}$

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Answer

We know that,
$\text{x}^{\text{n}}\text{ dx}-\frac{\text{x}^{\text{n}-1}}{\text{n}+1}+\text{C}$
Now,
$\int_{4}^\limits{9}\frac{1}{\sqrt{\text{x}}}\text{ dx}$
$=\Bigg[\frac{\text{x}^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}\Bigg]^9_4$
$=\Bigg[\frac{\sqrt{\text{x}}}{\frac{1}{2}}\Bigg]^9_4$
$=2\big[\sqrt{9}-\sqrt{4}\big]$
$=2\big[3-2\big]$
$=2$

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