NCERT Class 12 · Ch 7 Exercise 7.7 · Q10
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.7 Question 10
Evaluate \(\int \sqrt{x^{2} + 1}\, dx\).
💡 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: A: \(\frac{x \sqrt{x^{2} + 1}}{2} + \frac{\log{\left(\left|{x + \sqrt{x^{2} + 1}}\right| \right)}}{2}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Use the positive-quadratic root formula with A=1 (obtained by the integration-by-parts identity above).
- An antiderivative is F(x)=\(\frac{x \sqrt{x^{2} + 1}}{2} + \frac{\log{\left(\left|{x + \sqrt{x^{2} + 1}}\right| \right)}}{2}\).
- Differentiate this expression to check that F′(x)=\(\sqrt{x^{2} + 1}\). Add an arbitrary constant C on each domain component.
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