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Q11:
Integrate the functions in Exercises 1 to 23.
11. $\frac{1}{9x^2+6x+5}$
Solution :
Step 1: Analyzing the Integral Structure
The given integral is of the form:
$I = \int \frac{1}{9x^2 + 6x + 5} \, dx$
To evaluate this, we must express the denominator as a sum of squares, following the standard form $\int \frac{1}{a^2 + u^2} \, du = \frac{1}{a} \tan^{-1}\left(\frac{u}{a}\right) + C$ [Per the standard integral table].
Step 2: Completing the Square for the Denominator
Identify the quadratic expression: $f(x) = 9x^2 + 6x + 5$.
- Factor out the coefficient of $x^2$: $9\left(x^2 + \frac{6}{9}x + \frac{5}{9}\right) = 9\left(x^2 + \frac{2}{3}x + \frac{5}{9}\right)$.
- To complete the square, add and subtract the square of half the coefficient of $x$: $\left(\frac{1}{2} \cdot \frac{2}{3}\right)^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9}$.
- Rewrite the expression: $9\left[\left(x^2 + \frac{2}{3}x + \frac{1}{9}\right) + \frac{5}{9} - \frac{1}{9}\right]$.
- Simplify: $9\left[\left(x + \frac{1}{3}\right)^2 + \frac{4}{9}\right]$.
Step 3: Transforming the Integral
Substitute the completed square back into the integral:
$I = \int \frac{1}{9\left[\left(x + \frac{1}{3}\right)^2 + \left(\frac{2}{3}\right)^2\right]} \, dx$
$I = \frac{1}{9} \int \frac{1}{\left(x + \frac{1}{3}\right)^2 + \left(\frac{2}{3}\right)^2} \, dx$
Step 4: Applying the Standard Formula
Using the formula $\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C$, where $u = x + \frac{1}{3}$ and $a = \frac{2}{3}$:
$I = \frac{1}{9} \left[ \frac{1}{2/3} \tan^{-1}\left( \frac{x + 1/3}{2/3} \right) \right] + C$
$I = \frac{1}{9} \cdot \frac{3}{2} \tan^{-1}\left( \frac{\frac{3x+1}{3}}{\frac{2}{3}} \right) + C$
Step 5: Final Simplification
$I = \frac{3}{18} \tan^{-1}\left( \frac{3x+1}{2} \right) + C$
$I = \frac{1}{6} \tan^{-1}\left( \frac{3x+1}{2} \right) + C$
Final Solution: $\frac{1}{6} \tan^{-1}\left( \frac{3x+1}{2} \right) + C$
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. $\frac{1}{\sqrt{x^2+2x+2}}$
- 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. $\frac{x+2}{\sqrt{4x-x^2}}$
- 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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