NCERT Class 12 · Ch 7 Exercise 7.2 · Q10
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.2 Question 10
Evaluate \(\int \frac{1}{- \sqrt{x} + x}\, dx\).
💡 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: \(2 \log{\left(\left|{\sqrt{x} - 1}\right| \right)}\) + C.
- Work on each open interval where the original integrand is real and defined.
- u=√x, dx=2u du; the integrand becomes 2du/(u−1), for x>0,x≠1.
- An antiderivative is F(x)=\(2 \log{\left(\left|{\sqrt{x} - 1}\right| \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{- \sqrt{x} + x}\). Add an arbitrary constant C on each domain component.
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