NCERT Class 12 · Ch 7 Exercise 7.2 · Q23
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.2 Question 23
Evaluate \(\int \frac{\operatorname{asin}{\left(x \right)}}{\sqrt{1 - x^{2}}}\, dx\).
💡 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: \(\frac{\operatorname{asin}^{2}{\left(x \right)}}{2}\) + C.
- Work on each open interval where the original integrand is real and defined.
- u=sin⁻¹x, du=dx/√(1−x²), −1<x<1.
- An antiderivative is F(x)=\(\frac{\operatorname{asin}^{2}{\left(x \right)}}{2}\).
- Differentiate this expression to check that F′(x)=\(\frac{\operatorname{asin}{\left(x \right)}}{\sqrt{1 - x^{2}}}\). Add an arbitrary constant C on each domain component.
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