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Q9:
Integrate the functions in Exercises 1 to 9.
9. $\sqrt{1+\frac{x^2}{9}}$
Solution :
Step 1: Simplify the Integral Expression
The given integral is $\int \sqrt{1 + \frac{x^2}{9}} \, dx$. First, we rewrite the expression under the square root to facilitate the application of standard integration formulas: $$\sqrt{1 + \frac{x^2}{9}} = \sqrt{\frac{9 + x^2}{9}} = \frac{1}{3} \sqrt{3^2 + x^2}$$ [By algebraic simplification of the radicand].
Step 2: Selecting the Appropriate Integration Formula
The integral now takes the form $\frac{1}{3} \int \sqrt{a^2 + x^2} \, dx$, where $a = 3$. We utilize the standard integral identity for the square root of the sum of squares: $$\int \sqrt{a^2 + x^2} \, dx = \frac{x}{2} \sqrt{a^2 + x^2} + \frac{a^2}{2} \ln \left| x + \sqrt{a^2 + x^2} \right| + C$$ [Per the integration formula for $\sqrt{a^2 + x^2}$].
Step 3: Substituting Variables into the Formula
Substitute $a = 3$ into the identity:
$$I = \frac{1}{3} \left[ \frac{x}{2} \sqrt{3^2 + x^2} + \frac{3^2}{2} \ln \left| x + \sqrt{3^2 + x^2} \right| \right] + C$$ $$I = \frac{1}{3} \left[ \frac{x}{2} \sqrt{9 + x^2} + \frac{9}{2} \ln \left| x + \sqrt{9 + x^2} \right| \right] + C$$Step 4: Algebraic Distribution and Final Simplification
Distribute the factor $\frac{1}{3}$ across the bracketed terms:
$$I = \frac{x}{6} \sqrt{9 + x^2} + \frac{9}{6} \ln \left| x + \sqrt{9 + x^2} \right| + C$$ $$I = \frac{x}{6} \sqrt{9 + x^2} + \frac{3}{2} \ln \left| x + \sqrt{9 + x^2} \right| + C$$To align the result with the original form $\sqrt{1 + \frac{x^2}{9}}$, note that $\sqrt{9+x^2} = \sqrt{9(1 + \frac{x^2}{9})} = 3\sqrt{1 + \frac{x^2}{9}}$:
$$I = \frac{x}{6} \cdot 3\sqrt{1 + \frac{x^2}{9}} + \frac{3}{2} \ln \left| x + 3\sqrt{1 + \frac{x^2}{9}} \right| + C$$ $$I = \frac{x}{2} \sqrt{1 + \frac{x^2}{9}} + \frac{3}{2} \ln \left| x + 3\sqrt{1 + \frac{x^2}{9}} \right| + C$$Final Solution: \frac{x}{6} \sqrt{9+x^2} + \frac{3}{2} \ln \left| x + \sqrt{9+x^2} \right| + C
More Questions from Class 12 Mathematics Integrals EXERCISE 7.7
- Q1: Integrate the functions in Exercises 1 to 9. 1. $\sqrt{4-x^2}$
- Q2: Integrate the functions in Exercises 1 to 9. 2. $\sqrt{1-4x^2}$
- Q3: Integrate the functions in Exercises 1 to 9. 3. $\sqrt{x^2+4x+6}$
- Q4: Integrate the functions in Exercises 1 to 9. 4. $\sqrt{x^2+4x+1}$
- Q5: Integrate the functions in Exercises 1 to 9. 5. $\sqrt{1-4x-x^2}$
- Q6: Integrate the functions in Exercises 1 to 9. 6. $\sqrt{x^2+4x-5}$
- Q7: Integrate the functions in Exercises 1 to 9. 7. $\sqrt{1+3x-x^2}$
- Q8: Integrate the functions in Exercises 1 to 9. 8. $\sqrt{x^2+3x}$
- Q10: Choose the correct answer in Exercises 10 to 11. 10. $\int \sqrt{1+x^2} dx$ is equal to
- Q11: Choose the correct answer in Exercises 10 to 11. 11. $\int \sqrt{x^2-8x+7} dx$ is equal to
CBSE Solutions for Class 12 Mathematics Integrals
- Integrals Class 12 EXERCISE 7.1
- Integrals Class 12 EXERCISE 7.10
- Integrals Class 12 EXERCISE 7.2
- Integrals Class 12 EXERCISE 7.3
- Integrals Class 12 EXERCISE 7.4
- Integrals Class 12 EXERCISE 7.5
- Integrals Class 12 EXERCISE 7.6
- Integrals Class 12 EXERCISE 7.8
- Integrals Class 12 EXERCISE 7.9
- Integrals Class 12 Miscellaneous Exercise on Chapter 7
Chapters in CBSE - Class 12 Mathematics
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