NCERT Class 12 · Ch 7 Exercise 7.8 · Q22
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.8 Question 22
Evaluate \(\int\limits_{0}^{\frac{2}{3}} \frac{1}{9 x^{2} + 4}\, dx\).
💡 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: C: \(\frac{\pi}{24}\)
- Work on each open interval where the original integrand is real and defined.
- Set u=3x/2, so the primitive is tan⁻¹(3x/2)/6.
- An antiderivative is F(x)=\(\frac{\operatorname{atan}{\left(\frac{3 x}{2} \right)}}{6}\).
- At the upper and lower limits, F(b)=\(\frac{\pi}{24}\) and F(a)=\(0\).
- By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(\frac{\pi}{24}\).
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