NCERT Class 12 · Ch 7 Exercise 7.9 · Q3
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.9 Question 3
Evaluate \(\int\limits_{0}^{1} \operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)}\, dx\).
💡 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: \(- \log{\left(2 \right)} + \frac{\pi}{2}\)
- Work on each open interval where the original integrand is real and defined.
- On [0,1], principal sin⁻¹(2x/(1+x²))=2tan⁻¹x. Integrate by parts.
- An antiderivative is F(x)=\(2 x \operatorname{atan}{\left(x \right)} - \log{\left(x^{2} + 1 \right)}\).
- At the upper and lower limits, F(b)=\(- \log{\left(2 \right)} + \frac{\pi}{2}\) and F(a)=\(0\).
- By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(- \log{\left(2 \right)} + \frac{\pi}{2}\).
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