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Q2:
Integrate the rational functions in Exercises 1 to 21.
2. $\frac{1}{x^2-9}$
Solution :
Given Variables & Initial Setup
We are required to evaluate the integral of the rational function:
$I = \int \frac{1}{x^2 - 9} \, dx$
This integral takes the form $\int \frac{1}{x^2 - a^2} \, dx$, where $a = 3$. We shall employ the Method of Partial Fractions to decompose the integrand.
Step 1: Factorizing the Denominator
Using the algebraic identity for the difference of squares, $a^2 - b^2 = (a - b)(a + b)$, we factor the denominator:
$x^2 - 9 = (x - 3)(x + 3)$ [By Algebraic Identity $a^2 - b^2$]
Step 2: Partial Fraction Decomposition
We express the integrand as a sum of partial fractions:
$\frac{1}{(x - 3)(x + 3)} = \frac{A}{x - 3} + \frac{B}{x + 3}$
Multiplying both sides by $(x - 3)(x + 3)$ yields:
$1 = A(x + 3) + B(x - 3)$
Step 3: Solving for Coefficients A and B
To find $A$ and $B$, we equate the coefficients or substitute convenient values for $x$:
- Let $x = 3$: $1 = A(3 + 3) + B(0) \implies 6A = 1 \implies A = \frac{1}{6}$
- Let $x = -3$: $1 = A(0) + B(-3 - 3) \implies -6B = 1 \implies B = -\frac{1}{6}$
Step 4: Integration
Substituting the values of $A$ and $B$ back into the integral:
$I = \int \left( \frac{1/6}{x - 3} - \frac{1/6}{x + 3} \right) \, dx$
$I = \frac{1}{6} \int \frac{1}{x - 3} \, dx - \frac{1}{6} \int \frac{1}{x + 3} \, dx$
Using the standard integral formula $\int \frac{1}{u} \, du = \ln|u| + C$ [By Integration Principle of Linear Forms]:
$I = \frac{1}{6} \ln|x - 3| - \frac{1}{6} \ln|x + 3| + C$
Step 5: Applying Logarithmic Properties
Using the property $\ln(m) - \ln(n) = \ln\left(\frac{m}{n}\right)$:
$I = \frac{1}{6} \ln\left| \frac{x - 3}{x + 3} \right| + C$
Final Solution: $\frac{1}{6} \ln\left| \frac{x - 3}{x + 3} \right| + C$
More Questions from Class 12 Mathematics Integrals EXERCISE 7.5
- Q1: Integrate the rational functions in Exercises 1 to 21. 1. $\frac{x}{(x+1)(x+2)}$
- 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. $\frac{2}{(1-x)(1+x^2)}$
- 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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