NCERT Class 12 · Ch 7 Exercise 7.4 · Q13
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.4 Question 13
Evaluate \(\int \frac{1}{\sqrt{\left(x - 2\right) \left(x - 1\right)}}\, dx\).
💡 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: \(\log{\left(\left|{x + \sqrt{\left(x - 2\right) \left(x - 1\right)} - \frac{3}{2}}\right| \right)}\) + C.
- Work on each open interval where the integrand is real and defined.
- Complete the square: (x−3/2)²−1/4. Use the logarithmic standard form.
- An antiderivative is F(x)=\(\log{\left(\left|{x + \sqrt{\left(x - 2\right) \left(x - 1\right)} - \frac{3}{2}}\right| \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{\sqrt{\left(x - 2\right) \left(x - 1\right)}}\). Add an arbitrary constant C on each domain component.
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