NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q28
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 28
Evaluate \(\int\limits_{0}^{1} \frac{1}{- \sqrt{x} + \sqrt{x + 1}}\, dx\).
💡 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: \(\frac{4 \sqrt{2}}{3}\)
- Work on each open interval where the original integrand is real and defined.
- Rationalize: 1/(√(1+x)−√x)=√(1+x)+√x.
- An antiderivative is F(x)=\(\frac{2 x^{\frac{3}{2}}}{3} + \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3}\).
- At the upper and lower limits, F(b)=\(\frac{2}{3} + \frac{4 \sqrt{2}}{3}\) and F(a)=\(\frac{2}{3}\).
- By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(\frac{4 \sqrt{2}}{3}\).
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