NCERT Class 12 · Ch 7 Exercise 7.4 · Q7
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.4 Question 7
Evaluate \(\int \frac{x - 1}{\sqrt{x^{2} - 1}}\, dx\).
💡 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: \(\sqrt{x^{2} - 1} - \log{\left(\left|{x + \sqrt{x^{2} - 1}}\right| \right)}\) + C.
- Work on each open interval where the integrand is real and defined.
- Split into x/√(x²−1) minus 1/√(x²−1).
- An antiderivative is F(x)=\(\sqrt{x^{2} - 1} - \log{\left(\left|{x + \sqrt{x^{2} - 1}}\right| \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{x - 1}{\sqrt{x^{2} - 1}}\). Add an arbitrary constant C on each domain component.
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