Choose the correct answer in Exercise: ^ 3 _ 1 dx 1+x^ 2 equals 1… - Vidyadip

Question

Choose the correct answer in Exercise:
$\int^{\sqrt{3}}_{1}\frac{\text{dx}}{1+\text{x}^{2}}\text{equals}$

$\frac{\pi}{3}$
$\frac{2\pi}{3}$
$\frac{\pi}{6}$
$\frac{\pi}{12}$

✓

Answer

$\frac{\pi}{12}$

$\int\frac{\text{dx}}{1+\text{x}^{2}}=\tan^{-1}\text{x}=\text{F}\text{(x)}$
By second fundamental theorem of calculus, we obtain
$\int\limits_{1}^{\sqrt{3}}\frac{\text{dx}}{1+\text{x}^{2}}=\text{F}(\sqrt{3})-\text{F}(1)$
$=\tan^{-1}\sqrt{3}-\tan^{-1}1$
$=\frac{\pi}{3}-\frac{\pi}{4}$
$=\frac{\pi}{12}$

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