NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q19
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 19
Evaluate \(\int \sqrt{\frac{1 - \sqrt{x}}{\sqrt{x} + 1}}\, dx\).
💡 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: \(\sqrt{1 - x} \left(\sqrt{x} - 2\right) - \operatorname{asin}{\left(\sqrt{x} \right)}\) + C.
- Work for 0<x<1, with continuous primitive extension to the endpoints.
- Put u=√x∈(0,1). Then dx=2u du and √[(1−u)/(1+u)]=(1−u)/√(1−u²).
- Integrate 2u(1−u)/√(1−u²)du using the standard root/arcsine forms.
- An antiderivative is F(x)=\(\sqrt{1 - x} \left(\sqrt{x} - 2\right) - \operatorname{asin}{\left(\sqrt{x} \right)}\).
- Differentiate this expression to check that F′(x)=\(\sqrt{\frac{1 - \sqrt{x}}{\sqrt{x} + 1}}\). Add an arbitrary constant C on each domain component.
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