NCERT Class 12 · Ch 7 Exercise 7.8 · Q21
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.8 Question 21
Evaluate \(\int\limits_{1}^{\sqrt{3}} \frac{1}{x^{2} + 1}\, dx\).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: D: \(\frac{\pi}{12}\)
- Work on each open interval where the original integrand is real and defined.
- Use F=tan⁻¹x.
- An antiderivative is F(x)=\(\operatorname{atan}{\left(x \right)}\).
- At the upper and lower limits, F(b)=\(\frac{\pi}{3}\) and F(a)=\(\frac{\pi}{4}\).
- By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(\frac{\pi}{12}\).
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