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Q12:
Integrate the rational functions in Exercises 1 to 21.
12. $\frac{x^3+x+1}{x^2-1}$
Solution :
Step 1: Analyzing the Degree of the Rational Function
Given the integrand $f(x) = \frac{x^3 + x + 1}{x^2 - 1}$. Observe that the degree of the numerator $P(x) = x^3 + x + 1$ is 3, and the degree of the denominator $Q(x) = x^2 - 1$ is 2. Since the degree of the numerator is greater than or equal to the degree of the denominator, we must perform polynomial long division to express the function in a simpler form [Per the Division Algorithm for Polynomials].
Step 2: Polynomial Long Division
Divide $x^3 + x + 1$ by $x^2 - 1$:
| $x^3 + 0x^2 + x + 1$ divided by $x^2 - 1$ |
| Quotient: $x$ |
| Remainder: $(x^3 + x + 1) - x(x^2 - 1) = x^3 + x + 1 - x^3 + x = 2x + 1$ |
Thus, the expression can be rewritten as:
$\frac{x^3 + x + 1}{x^2 - 1} = x + \frac{2x + 1}{x^2 - 1}$
Step 3: Partial Fraction Decomposition
We decompose the proper fraction $\frac{2x + 1}{x^2 - 1}$ using partial fractions. Note that $x^2 - 1 = (x - 1)(x + 1)$.
$\frac{2x + 1}{(x - 1)(x + 1)} = \frac{A}{x - 1} + \frac{B}{x + 1}$
Multiplying by the common denominator $(x - 1)(x + 1)$:
$2x + 1 = A(x + 1) + B(x - 1)$
- Set $x = 1$: $2(1) + 1 = A(1 + 1) \implies 3 = 2A \implies A = \frac{3}{2}$
- Set $x = -1$: $2(-1) + 1 = B(-1 - 1) \implies -1 = -2B \implies B = \frac{1}{2}$
Step 4: Integration of the Components
Substitute the partial fractions back into the integral:
$\int \frac{x^3 + x + 1}{x^2 - 1} dx = \int \left( x + \frac{3/2}{x - 1} + \frac{1/2}{x + 1} \right) dx$
Apply the linearity property of integrals [Per the Sum Rule of Integration]:
$= \int x dx + \frac{3}{2} \int \frac{1}{x - 1} dx + \frac{1}{2} \int \frac{1}{x + 1} dx$
Perform the integration:
$= \frac{x^2}{2} + \frac{3}{2} \ln|x - 1| + \frac{1}{2} \ln|x + 1| + C$
Visual Representation of the integrand behavior near asymptotes
Final Solution: $\frac{x^2}{2} + \frac{3}{2} \ln|x - 1| + \frac{1}{2} \ln|x + 1| + C$
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. $\frac{5x}{(x+1)(x^2-4)}$
- 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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