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Q13:
Integrate the rational functions in Exercises 1 to 21.
13. $\frac{2}{(1-x)(1+x^2)}$
Solution :
Step 1: Decomposition via Partial Fractions
To integrate the rational function $f(x) = \frac{2}{(1-x)(1+x^2)}$, we express the integrand as a sum of partial fractions. Since $(1+x^2)$ is an irreducible quadratic factor, the partial fraction decomposition takes the form:
$\frac{2}{(1-x)(1+x^2)} = \frac{A}{1-x} + \frac{Bx + C}{1+x^2}$ [By the Theory of Partial Fraction Decomposition for distinct linear and irreducible quadratic factors]
Step 2: Determining the Constants
Equating the numerators after finding a common denominator:
$2 = A(1+x^2) + (Bx + C)(1-x)$
$2 = A + Ax^2 + Bx - Bx^2 + C - Cx$
$2 = (A - B)x^2 + (B - C)x + (A + C)$
By comparing coefficients of powers of $x$ on both sides:
- For $x^2$: $A - B = 0 \implies A = B$
- For $x^1$: $B - C = 0 \implies B = C$
- For $x^0$: $A + C = 2$
Substituting $A=B$ and $B=C$ into the constant term equation:
$A + A = 2 \implies 2A = 2 \implies A = 1$
Therefore, $A = 1, B = 1, C = 1$.
Step 3: Setting up the Integral
The integral becomes:
$\int \frac{2}{(1-x)(1+x^2)} dx = \int \frac{1}{1-x} dx + \int \frac{x+1}{1+x^2} dx$
Splitting the second integral:
$I = \int \frac{1}{1-x} dx + \int \frac{x}{1+x^2} dx + \int \frac{1}{1+x^2} dx$
Step 4: Evaluating the Components
- $\int \frac{1}{1-x} dx = -\ln|1-x| + C_1$ [Using substitution $u = 1-x, du = -dx$]
- $\int \frac{x}{1+x^2} dx = \frac{1}{2} \ln(1+x^2) + C_2$ [Using substitution $v = 1+x^2, dv = 2x dx$]
- $\int \frac{1}{1+x^2} dx = \tan^{-1}(x) + C_3$ [Per the standard integral formula for $\int \frac{1}{1+x^2}dx$]
Visual Representation of the Integrand
Final Synthesis
Combining the results and adding the constant of integration $C$:
$I = -\ln|1-x| + \frac{1}{2} \ln(1+x^2) + \tan^{-1}(x) + C$
Final Solution: -\ln|1-x| + \frac{1}{2} \ln(1+x^2) + \tan^{-1}(x) + 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. $\frac{x^3+x+1}{x^2-1}$
- 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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