_ 1 ^ 3 d x 1+x^ 2 equals - Vidyadip

Question

$\int_{1}^{\sqrt{3}} \frac{d x}{1+x^{2}}$ equals

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Answer

Let $I=\int_{1}^{\sqrt{3}} \frac{d x}{x^{2}+1}$
$\Rightarrow I=\int_{1}^{\sqrt{3}} \frac{d x}{x^{2}+1}$
$\Rightarrow I=\left[\tan ^{-1} x\right]_{1}^{\sqrt{3}}$ = $\left[\tan ^{-1} \sqrt{3}-\tan ^{-1} 1\right]$ = $\frac{\pi}{3}-\frac{\pi}{4}$ ...[$\because~\int \frac{d x}{a^{2}+x^{2}}=\frac{1}{a} \tan ^{-1} \frac{x}{a}+c$]
$\Rightarrow I=\frac{4 \pi-3 \pi}{12}=\frac{\pi}{12}$
$\therefore \int_{1}^{\sqrt{3}} \frac{d x}{x^{2}+1}=\frac{\pi}{12}$

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