NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q37
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 37
Evaluate \(\int\limits_{0}^{1} \operatorname{asin}{\left(x \right)}\, dx\).
💡 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: \(-1 + \frac{\pi}{2}\)
- Work on each open interval where the original integrand is real and defined.
- Integrate by parts with u=sin⁻¹x, dv=dx; use endpoint limits.
- An antiderivative is F(x)=\(x \operatorname{asin}{\left(x \right)} + \sqrt{1 - x^{2}}\).
- At the upper and lower limits, F(b)=\(\frac{\pi}{2}\) and F(a)=\(1\).
- By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(-1 + \frac{\pi}{2}\).
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