NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q2
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 2
Evaluate \(\int \frac{1}{\sqrt{a + x} + \sqrt{b + x}}\, dx\).
💡 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: For a≠b: 2[(x+a)^(3/2)−(x+b)^(3/2)]/[3(a−b)]+C. For a=b,x+a>0: √(x+a)+C. Keep the original real domain.
- Work on each open interval where the original integrand is real and defined.
- Rationalize to [√(x+a)−√(x+b)]/(a−b), a≠b. If a=b the primitive is √(x+a).
- An antiderivative is F(x)=\(\frac{2 \left(a + x\right)^{\frac{3}{2}} - 2 \left(b + x\right)^{\frac{3}{2}}}{3 a - 3 b}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{\sqrt{a + x} + \sqrt{b + x}}\). Add an arbitrary constant C on each domain component.
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