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

MCQ

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

A
$\frac{\pi}{3}$
✓
$\frac{2\pi}{3}$
C
$\frac{\pi}{6}$
D
$\frac{\pi}{12}$

✓

Answer

Correct option: B.

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

$\int\frac{\text{dx}}{1+\text{x}^{2}}=\tan^{-1}\text{x}=\text{F}\text{(x)}$

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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