NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q22
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 22
Evaluate \(\int \operatorname{atan}{\left(\sqrt{\frac{1 - x}{x + 1}} \right)}\, dx\).
💡 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: \(\frac{x \operatorname{acos}{\left(x \right)}}{2} - \frac{\sqrt{1 - x^{2}}}{2}\) + C.
- Work for −1<x<1.
- Put x=cos θ, 0<θ<π; the principal arctangent of √[(1−x)/(1+x)] is θ/2=cos⁻¹x/2.
- Integrate cos⁻¹x/2 by parts.
- An antiderivative is F(x)=\(\frac{x \operatorname{acos}{\left(x \right)}}{2} - \frac{\sqrt{1 - x^{2}}}{2}\).
- Differentiate this expression to check that F′(x)=\(\operatorname{atan}{\left(\sqrt{\frac{1 - x}{x + 1}} \right)}\). Add an arbitrary constant C on each domain component.
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