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Q5:
Integrate the rational functions in Exercises 1 to 21.
5. $\frac{2x}{x^2+3x+2}$
Solution :
Step 1: Factorization of the Denominator
To perform partial fraction decomposition, the denominator $x^2 + 3x + 2$ must be expressed as a product of linear factors. We seek two numbers that multiply to $2$ and add to $3$. These are $1$ and $2$.
$x^2 + 3x + 2 = (x + 1)(x + 2)$ [By Factorization of Quadratic Trinomials]
Step 2: Partial Fraction Decomposition
We represent the integrand as a sum of partial fractions:
$\frac{2x}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x+2}$
Equating the numerators:
$2x = A(x + 2) + B(x + 1)$
Step 3: Determining Constants A and B
To find the values of the constants, we apply the substitution method:
- Let $x = -1$: $2(-1) = A(-1 + 2) + B(-1 + 1)$ $-2 = A(1) \implies A = -2$
- Let $x = -2$: $2(-2) = A(-2 + 2) + B(-2 + 1)$ $-4 = B(-1) \implies B = 4$
Step 4: Integration
Substitute the constants back into the integral expression:
$\int \frac{2x}{(x+1)(x+2)} \, dx = \int \left( \frac{-2}{x+1} + \frac{4}{x+2} \right) \, dx$
[By Linearity of Integration]
$= -2 \int \frac{1}{x+1} \, dx + 4 \int \frac{1}{x+2} \, dx$
Applying the standard integral formula $\int \frac{1}{u} \, du = \ln|u| + C$:
$= -2 \ln|x+1| + 4 \ln|x+2| + C$
Step 5: Logarithmic Simplification
Using the properties of logarithms, specifically $n \ln(a) = \ln(a^n)$ and $\ln(a) - \ln(b) = \ln(\frac{a}{b})$:
$= \ln|x+2|^4 - \ln|x+1|^2 + C$
$= \ln \left| \frac{(x+2)^4}{(x+1)^2} \right| + C$
Final Solution: $-2 \ln|x+1| + 4 \ln|x+2| + C$ or $\ln \left| \frac{(x+2)^4}{(x+1)^2} \right| + C$
More Questions from Class 12 Mathematics Integrals EXERCISE 7.5
- Q1: Integrate the rational functions in Exercises 1 to 21. 1. $\frac{x}{(x+1)(x+2)}$
- 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. $\frac{x}{(x-1)(x-2)(x-3)}$
- 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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