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Q14:
Integrate the rational functions in Exercises 1 to 21.
14. $\frac{3x-1}{(x+2)^2}$
Solution :
Step 1: Decomposition into Partial Fractions
To integrate the rational function $I = \int \frac{3x-1}{(x+2)^2} \, dx$, we express the integrand as a sum of partial fractions. For a repeated linear factor in the denominator, the decomposition is given by:
$\frac{3x-1}{(x+2)^2} = \frac{A}{x+2} + \frac{B}{(x+2)^2}$ [By Partial Fraction Decomposition Theorem for repeated roots]
Step 2: Determining Constants A and B
Multiply both sides by $(x+2)^2$ to clear the denominators:
$3x - 1 = A(x + 2) + B$
$3x - 1 = Ax + 2A + B$
By comparing the coefficients of like powers of $x$ on both sides:
- Coefficient of $x$: $A = 3$
- Constant term: $2A + B = -1$
Substitute $A = 3$ into the constant term equation:
$2(3) + B = -1 \implies 6 + B = -1 \implies B = -7$
Step 3: Setting up the Integral
Substitute the constants $A$ and $B$ back into the integral expression:
$I = \int \left( \frac{3}{x+2} - \frac{7}{(x+2)^2} \right) \, dx$
$I = 3 \int \frac{1}{x+2} \, dx - 7 \int (x+2)^{-2} \, dx$
Step 4: Execution of Integration
Applying the standard integration rules:
- $\int \frac{1}{x} \, dx = \ln|x| + C$ [Per the logarithmic integration rule]
- $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$ [Per the Power Rule for Integration]
$I = 3 \ln|x+2| - 7 \left( \frac{(x+2)^{-2+1}}{-2+1} \right) + C$
$I = 3 \ln|x+2| - 7 \left( \frac{(x+2)^{-1}}{-1} \right) + C$
$I = 3 \ln|x+2| + \frac{7}{x+2} + C$
Final Solution: $3 \ln|x+2| + \frac{7}{x+2} + C$
More Questions from Class 12 Mathematics Integrals EXERCISE 7.5
- Q1: Integrate the rational functions in Exercises 1 to 21. 1. $\frac{x}{(x+1)(x+2)}$
- Q2: Integrate the rational functions in Exercises 1 to 21. 2. $\frac{1}{x^2-9}$
- Q3: Integrate the rational functions in Exercises 1 to 21. 3. $\frac{3x-1}{(x-1)(x-2)(x-3)}$
- Q4: Integrate the rational functions in Exercises 1 to 21. 4. $\frac{x}{(x-1)(x-2)(x-3)}$
- Q5: Integrate the rational functions in Exercises 1 to 21. 5. $\frac{2x}{x^2+3x+2}$
- Q6: Integrate the rational functions in Exercises 1 to 21. 6. $\frac{1-x^2}{x(1-2x)}$
- Q7: Integrate the rational functions in Exercises 1 to 21. 7. $\frac{x}{(x^2+1)(x-1)}$
- Q8: Integrate the rational functions in Exercises 1 to 21. 8. $\frac{x}{(x-1)^2(x+2)}$
- Q9: Integrate the rational functions in Exercises 1 to 21. 9. $\frac{3x+5}{x^3-x^2-x+1}$
- Q10: Integrate the rational functions in Exercises 1 to 21. 10. $\frac{2x-3}{(x^2-1)(2x+3)}$
- Q11: Integrate the rational functions in Exercises 1 to 21. 11. $\frac{5x}{(x+1)(x^2-4)}$
- Q12: Integrate the rational functions in Exercises 1 to 21. 12. $\frac{x^3+x+1}{x^2-1}$
- Q13: Integrate the rational functions in Exercises 1 to 21. 13. $\frac{2}{(1-x)(1+x^2)}$
- Q15: Integrate the rational functions in Exercises 1 to 21. 15. $\frac{1}{x^4-1}$
- Q16: Integrate the rational functions in Exercises 1 to 21. 16. $\frac{1}{x(x^n+1)}$ [Hint: multiply numerator and denominator by $x^{n-1}$ and put $x^n = t$]
- Q17: Integrate the rational functions in Exercises 1 to 21. 17. $\frac{\cos x}{(1-\sin x)(2-\sin x)}$ [Hint : Put $\sin x = t$]
- Q18: Integrate the rational functions in Exercises 1 to 21. 18. $\frac{(x^2+1)(x^2+2)}{(x^2+3)(x^2+4)}$
- Q19: Integrate the rational functions in Exercises 1 to 21. 19. $\frac{2x}{(x^2+1)(x^2+3)}$
- Q20: Integrate the rational functions in Exercises 1 to 21. 20. $\frac{1}{x(x^4-1)}$
- Q21: Integrate the rational functions in Exercises 1 to 21. 21. $\frac{1}{e^x-1}$ [Hint : Put $e^x=t$]
- Q22: Choose the correct answer in each of the Exercises 22 and 23. 22. $\int \frac{x dx}{(x-1)(x-2)}$ equals
- Q23: Choose the correct answer in each of the Exercises 22 and 23. 23. $\int \frac{dx}{x(x^2+1)}$ 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 7.4
- 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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