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Q21:
Integrate the functions in Exercises 1 to 23.
21. $\frac{x+2}{\sqrt{x^2+2x+3}}$
Solution :
Step 1: Decomposition of the Integrand
We are required to evaluate the integral: $I = \int \frac{x+2}{\sqrt{x^2+2x+3}} \, dx$
To solve this, we express the numerator $(x+2)$ in terms of the derivative of the quadratic expression under the square root, $f(x) = x^2+2x+3$. The derivative is $f'(x) = 2x+2$.
We set: $x+2 = A \frac{d}{dx}(x^2+2x+3) + B$
$x+2 = A(2x+2) + B$
$x+2 = 2Ax + (2A + B)$
Comparing coefficients of $x$ and constants [By Principle of Equating Coefficients]:
- $2A = 1 \implies A = \frac{1}{2}$
- $2A + B = 2 \implies 2(\frac{1}{2}) + B = 2 \implies 1 + B = 2 \implies B = 1$
Step 2: Splitting the Integral
Rewrite the original integral using these constants:
$I = \int \frac{\frac{1}{2}(2x+2) + 1}{\sqrt{x^2+2x+3}} \, dx$ $I = \frac{1}{2} \int \frac{2x+2}{\sqrt{x^2+2x+3}} \, dx + \int \frac{1}{\sqrt{x^2+2x+3}} \, dx$Let $I = \frac{1}{2} I_1 + I_2$, where $I_1 = \int \frac{2x+2}{\sqrt{x^2+2x+3}} \, dx$ and $I_2 = \int \frac{1}{\sqrt{x^2+2x+3}} \, dx$.
Step 3: Solving $I_1$
For $I_1$, let $u = x^2+2x+3$, then $du = (2x+2)dx$ [By Method of Substitution].
$I_1 = \int u^{-1/2} \, du = 2u^{1/2} + C_1 = 2\sqrt{x^2+2x+3} + C_1$Step 4: Solving $I_2$
For $I_2$, complete the square for the denominator: $x^2+2x+3 = (x^2+2x+1) + 2 = (x+1)^2 + (\sqrt{2})^2$.
$I_2 = \int \frac{dx}{\sqrt{(x+1)^2 + (\sqrt{2})^2}}$Using the standard integration formula $\int \frac{dx}{\sqrt{x^2+a^2}} = \ln|x + \sqrt{x^2+a^2}| + C$:
$I_2 = \ln|(x+1) + \sqrt{(x+1)^2 + 2}| + C_2 = \ln|x+1 + \sqrt{x^2+2x+3}| + C_2$Step 5: Final Compilation
$I = \frac{1}{2} (2\sqrt{x^2+2x+3}) + \ln|x+1 + \sqrt{x^2+2x+3}| + C$
$I = \sqrt{x^2+2x+3} + \ln|x+1 + \sqrt{x^2+2x+3}| + C$
Final Solution: \sqrt{x^2+2x+3} + \ln|x+1 + \sqrt{x^2+2x+3}| + 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. $\frac{4x+1}{\sqrt{2x^2+x-3}}$
- Q17: Integrate the functions in Exercises 1 to 23. 17. $\frac{x+2}{\sqrt{x^2-1}}$
- 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. $\frac{x+2}{\sqrt{4x-x^2}}$
- 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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