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Q10:
Integrate the rational functions in Exercises 1 to 21.
10. $\frac{2x-3}{(x^2-1)(2x+3)}$
Solution :
Step 1: Factorization and Decomposition Setup
The given integrand is $I = \int \frac{2x-3}{(x^2-1)(2x+3)} \, dx$. First, factorize the quadratic term $(x^2-1)$ as $(x-1)(x+1)$ [By difference of squares identity $a^2-b^2 = (a-b)(a+b)$].
The expression becomes: $\frac{2x-3}{(x-1)(x+1)(2x+3)}$.
We decompose this into partial fractions as follows:
$\frac{2x-3}{(x-1)(x+1)(2x+3)} = \frac{A}{x-1} + \frac{B}{x+1} + \frac{C}{2x+3}$
Step 2: Determining Constants $A, B,$ and $C$
Multiplying both sides by the denominator $(x-1)(x+1)(2x+3)$, we obtain:
$2x-3 = A(x+1)(2x+3) + B(x-1)(2x+3) + C(x-1)(x+1)$
- For $A$ (Let $x = 1$): $2(1)-3 = A(1+1)(2(1)+3) \Rightarrow -1 = A(2)(5) \Rightarrow A = -\frac{1}{10}$
- For $B$ (Let $x = -1$): $2(-1)-3 = B(-1-1)(2(-1)+3) \Rightarrow -5 = B(-2)(1) \Rightarrow B = \frac{5}{2}$
- For $C$ (Let $x = -\frac{3}{2}$): $2(-\frac{3}{2})-3 = C(-\frac{3}{2}-1)(-\frac{3}{2}+1) \Rightarrow -6 = C(-\frac{5}{2})(-\frac{1}{2}) \Rightarrow -6 = C(\frac{5}{4}) \Rightarrow C = -\frac{24}{5}$
Step 3: Integration of Partial Fractions
Substituting the constants back into the integral expression:
$I = \int \left( -\frac{1}{10(x-1)} + \frac{5}{2(x+1)} - \frac{24}{5(2x+3)} \right) dx$
Applying the linearity property of integrals $\int (af(x) + bg(x)) dx = a \int f(x) dx + b \int g(x) dx$ and the standard integral $\int \frac{1}{ax+b} dx = \frac{1}{a} \ln|ax+b| + C$:
$I = -\frac{1}{10} \ln|x-1| + \frac{5}{2} \ln|x+1| - \frac{24}{5} \cdot \frac{1}{2} \ln|2x+3| + K$
$I = -\frac{1}{10} \ln|x-1| + \frac{5}{2} \ln|x+1| - \frac{12}{5} \ln|2x+3| + K$
Step 4: Geometric Representation of the Integrand
Final Solution: $-\frac{1}{10} \ln|x-1| + \frac{5}{2} \ln|x+1| - \frac{12}{5} \ln|2x+3| + K$
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. $\frac{3x+5}{x^3-x^2-x+1}$
- 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. $\frac{2}{(1-x)(1+x^2)}$
- Q14: Integrate the rational functions in Exercises 1 to 21. 14. $\frac{3x-1}{(x+2)^2}$
- 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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