$\displaystyle\int_1^{\sqrt{3}} \dfrac{dx}{1+x^2}$ equals
A$\dfrac{\pi}{3}$
B$\dfrac{2\pi}{3}$
C$\dfrac{\pi}{6}$
D$\dfrac{\pi}{12}$
Answer & Solution
Correct answer: D. $\dfrac{\pi}{12}$
1. Antiderivative: $\tan^{-1}x$.
2. Evaluate: $\tan^{-1}\sqrt{3}-\tan^{-1}1=\dfrac{\pi}{3}-\dfrac{\pi}{4}$.
3. $=\dfrac{4\pi-3\pi}{12}=\dfrac{\pi}{12}$.
4. Option A forgets to subtract $\tan^{-1}1$.
_Source: NCERT Class 12 Mathematics Ch 7 "Integrals", p.338_
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