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Q23:
Choose the correct answer in each of the Exercises 22 and 23.
23. $\int \frac{dx}{x(x^2+1)}$ equals
Solution :
Step 1: Decomposition via Partial Fractions
To integrate $I = \int \frac{1}{x(x^2+1)} \, dx$, we express the integrand as a sum of partial fractions. Since $x^2+1$ is an irreducible quadratic factor, the decomposition takes the following form:
$\frac{1}{x(x^2+1)} = \frac{A}{x} + \frac{Bx + C}{x^2+1}$ [By the Method of Partial Fractions for irreducible quadratic factors]
Step 2: Solving for Coefficients A, B, and C
Multiply both sides by the denominator $x(x^2+1)$:
$1 = A(x^2+1) + (Bx+C)x$
$1 = Ax^2 + A + Bx^2 + Cx$
$1 = (A+B)x^2 + Cx + A$
By comparing the coefficients of the powers of $x$ on both sides:
- Constant term: $A = 1$
- Coefficient of $x$: $C = 0$
- Coefficient of $x^2$: $A+B = 0 \implies 1+B = 0 \implies B = -1$
Step 3: Rewriting the Integral
Substitute the constants back into the partial fraction decomposition:
$I = \int \left( \frac{1}{x} + \frac{-x + 0}{x^2+1} \right) \, dx$
$I = \int \frac{1}{x} \, dx - \int \frac{x}{x^2+1} \, dx$
Step 4: Execution of Integration
For the first term, $\int \frac{1}{x} \, dx = \ln|x|$.
For the second term, use the method of substitution. Let $u = x^2 + 1$, then $du = 2x \, dx$, which implies $x \, dx = \frac{1}{2} du$.
$I = \ln|x| - \frac{1}{2} \int \frac{1}{u} \, du$ [By the power rule for integration of reciprocals]
$I = \ln|x| - \frac{1}{2} \ln|x^2+1| + C$
Step 5: Logarithmic Simplification
Using the property $a \ln(b) = \ln(b^a)$:
$I = \ln|x| - \ln(x^2+1)^{1/2} + C$
$I = \ln \left| \frac{x}{\sqrt{x^2+1}} \right| + C$
Final Solution: \log|x| - \frac{1}{2}\log(x^2+1) + C = \log \left| \frac{x}{\sqrt{x^2+1}} \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. $\frac{x}{(x-1)(x-2)(x-3)}$
- 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
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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