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Q17:
Integrate the functions in Exercises 1 to 23.
17. $\frac{x+2}{\sqrt{x^2-1}}$
Solution :
Given Variables & Initial Setup
We are required to evaluate the indefinite integral:
$I = \int \frac{x+2}{\sqrt{x^2-1}} \, dx$
Step 1: Decomposition of the Integrand
To facilitate integration, we split the numerator to align with the derivative of the radicand ($x^2 - 1$). The derivative of $x^2 - 1$ is $2x$. We rewrite the numerator $x + 2$ as follows:
$x + 2 = \frac{1}{2}(2x) + 2$
[By the method of splitting the numerator for integrals of the form $\int \frac{px+q}{\sqrt{ax^2+bx+c}} dx$]
Step 2: Splitting the Integral
Substitute the decomposition into the integral:
$I = \int \frac{\frac{1}{2}(2x) + 2}{\sqrt{x^2-1}} \, dx$
$I = \frac{1}{2} \int \frac{2x}{\sqrt{x^2-1}} \, dx + 2 \int \frac{1}{\sqrt{x^2-1}} \, dx$
Let $I = \frac{1}{2}I_1 + 2I_2$, where $I_1 = \int \frac{2x}{\sqrt{x^2-1}} \, dx$ and $I_2 = \int \frac{1}{\sqrt{x^2-1}} \, dx$.
Step 3: Evaluating $I_1$ via Substitution
For $I_1 = \int \frac{2x}{\sqrt{x^2-1}} \, dx$:
Let $u = x^2 - 1$. Then $du = 2x \, dx$. [By Method of Substitution]
$I_1 = \int \frac{du}{\sqrt{u}} = \int u^{-1/2} \, du$
$I_1 = \frac{u^{1/2}}{1/2} + C_1 = 2\sqrt{x^2-1} + C_1$
Step 4: Evaluating $I_2$ via Standard Integral Formula
For $I_2 = \int \frac{1}{\sqrt{x^2-1}} \, dx$:
Using the standard integral identity: $\int \frac{1}{\sqrt{x^2-a^2}} \, dx = \ln|x + \sqrt{x^2-a^2}| + C$. [Standard Integral Formula No. 4]
$I_2 = \ln|x + \sqrt{x^2-1}| + C_2$
Step 5: Combining Results
Substitute the results of $I_1$ and $I_2$ back into the equation for $I$:
$I = \frac{1}{2} [2\sqrt{x^2-1}] + 2 [\ln|x + \sqrt{x^2-1}|] + C$
$I = \sqrt{x^2-1} + 2\ln|x + \sqrt{x^2-1}| + C$
Visual Representation of the Function
Final Solution: \sqrt{x^2-1} + 2\ln|x + \sqrt{x^2-1}| + C
More Questions from Class 12 Mathematics Integrals EXERCISE 7.4
- Q1: Integrate the functions in Exercises 1 to 23. 1. $\frac{3x^2}{x^6+1}$
- Q2: Integrate the functions in Exercises 1 to 23. 2. $\frac{1}{\sqrt{1+4x^2}}$
- Q3: Integrate the functions in Exercises 1 to 23. 3. $\frac{1}{\sqrt{(2-x)^2+1}}$
- Q4: Integrate the functions in Exercises 1 to 23. 4. $\frac{1}{\sqrt{9-25x^2}}$
- Q5: Integrate the functions in Exercises 1 to 23. 5. $\frac{3x}{1+2x^4}$
- Q6: Integrate the functions in Exercises 1 to 23. 6. $\frac{x^2}{1-x^6}$
- Q7: Integrate the functions in Exercises 1 to 23. 7. $\frac{x-1}{\sqrt{x^2-1}}$
- Q8: Integrate the functions in Exercises 1 to 23. 8. $\frac{x^2}{\sqrt{x^6+a^6}}$
- Q9: Integrate the functions in Exercises 1 to 23. 9. $\frac{\sec^2 x}{\sqrt{\tan^2 x + 4}}$
- Q10: Integrate the functions in Exercises 1 to 23. 10. $\frac{1}{\sqrt{x^2+2x+2}}$
- Q11: Integrate the functions in Exercises 1 to 23. 11. $\frac{1}{9x^2+6x+5}$
- Q12: Integrate the functions in Exercises 1 to 23. 12. $\frac{1}{\sqrt{7-6x-x^2}}$
- Q13: Integrate the functions in Exercises 1 to 23. 13. $\frac{1}{\sqrt{(x-1)(x-2)}}$
- Q14: Integrate the functions in Exercises 1 to 23. 14. $\frac{1}{\sqrt{8+3x-x^2}}$
- Q15: Integrate the functions in Exercises 1 to 23. 15. $\frac{1}{\sqrt{(x-a)(x-b)}}$
- Q16: Integrate the functions in Exercises 1 to 23. 16. $\frac{4x+1}{\sqrt{2x^2+x-3}}$
- Q18: Integrate the functions in Exercises 1 to 23. 18. $\frac{5x-2}{1+2x+3x^2}$
- Q19: Integrate the functions in Exercises 1 to 23. 19. $\frac{6x+7}{\sqrt{(x-5)(x-4)}}$
- Q20: Integrate the functions in Exercises 1 to 23. 20. $\frac{x+2}{\sqrt{4x-x^2}}$
- Q21: Integrate the functions in Exercises 1 to 23. 21. $\frac{x+2}{\sqrt{x^2+2x+3}}$
- Q22: Integrate the functions in Exercises 1 to 23. 22. $\frac{x+3}{x^2-2x-5}$
- Q23: Integrate the functions in Exercises 1 to 23. 23. $\frac{5x+3}{\sqrt{x^2+4x+10}}$
- Q24: Choose the correct answer in Exercises 24 and 25. 24. $\int \frac{dx}{x^2+2x+2}$ equals
- Q25: Choose the correct answer in Exercises 24 and 25. 25. $\int \frac{dx}{\sqrt{9x-4x^2}}$ equals
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 7.5
- Integrals Class 12 EXERCISE 7.6
- Integrals Class 12 EXERCISE 7.7
- 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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