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Q4:
Integrate the rational functions in Exercises 1 to 21.
4. $\frac{x}{(x-1)(x-2)(x-3)}$
Solution :
Step 1: Decomposition via Partial Fractions
To integrate the rational function $f(x) = \frac{x}{(x-1)(x-2)(x-3)}$, we express it as a sum of partial fractions [Per the Method of Partial Fraction Decomposition for distinct linear factors]:
$\frac{x}{(x-1)(x-2)(x-3)} = \frac{A}{x-1} + \frac{B}{x-2} + \frac{C}{x-3}$
Multiplying both sides by the denominator $(x-1)(x-2)(x-3)$, we obtain the identity:
$x = A(x-2)(x-3) + B(x-1)(x-3) + C(x-1)(x-2)$
Step 2: Determining Constants A, B, and C
We solve for the constants by substituting the roots of the denominator:
- For $x = 1$: $1 = A(1-2)(1-3) + B(0) + C(0) \implies 1 = A(-1)(-2) \implies 2A = 1 \implies A = \frac{1}{2}$
- For $x = 2$: $2 = A(0) + B(2-1)(2-3) + C(0) \implies 2 = B(1)(-1) \implies -B = 2 \implies B = -2$
- For $x = 3$: $3 = A(0) + B(0) + C(3-1)(3-2) \implies 3 = C(2)(1) \implies 2C = 3 \implies C = \frac{3}{2}$
Step 3: Integration of the Partial Fractions
Substitute the constants back into the integral expression:
$\int \frac{x}{(x-1)(x-2)(x-3)} dx = \int \left( \frac{1/2}{x-1} - \frac{2}{x-2} + \frac{3/2}{x-3} \right) dx$
Applying the linearity property of integrals [$\int [af(x) + bg(x)] dx = a \int f(x) dx + b \int g(x) dx$]:
$= \frac{1}{2} \int \frac{1}{x-1} dx - 2 \int \frac{1}{x-2} dx + \frac{3}{2} \int \frac{1}{x-3} dx$
Step 4: Evaluation and Final Result
Using the standard integral formula $\int \frac{1}{u} du = \ln|u| + C$:
$= \frac{1}{2} \ln|x-1| - 2 \ln|x-2| + \frac{3}{2} \ln|x-3| + C$
Final Solution: $\frac{1}{2} \ln|x-1| - 2 \ln|x-2| + \frac{3}{2} \ln|x-3| + C$
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. $\frac{3x-1}{(x-1)(x-2)(x-3)}$
- 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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