NCERT Class 12 · Ch 7 Exercise 7.7 · Q9
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.7 Question 9
Evaluate \(\int \sqrt{\frac{x^{2}}{9} + 1}\, dx\).
💡 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: \(\frac{x \sqrt{x^{2} + 9}}{6} + \frac{3 \log{\left(\left|{x + \sqrt{x^{2} + 9}}\right| \right)}}{2}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Write the integrand as √(x²+9)/3.
- For J=∫√(u²+A)du, integrate by parts with dv=du. Use u²=(u²+A)−A to get 2J=u√(u²+A)+A∫du/√(u²+A). For √(A−u²), the analogous identity uses an arcsine. Substitute the standard log or arcsine primitive.
- An antiderivative is F(x)=\(\frac{x \sqrt{x^{2} + 9}}{6} + \frac{3 \log{\left(\left|{x + \sqrt{x^{2} + 9}}\right| \right)}}{2}\).
- Differentiate this expression to check that F′(x)=\(\sqrt{\frac{x^{2}}{9} + 1}\). Add an arbitrary constant C on each domain component.
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