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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 $x^n = t$]
Solution :
Step 1: Application of the Algebraic Strategy
To integrate the function $I = \int \frac{1}{x(x^n+1)} \, dx$, we apply the suggested hint to facilitate substitution. Multiply the numerator and denominator by $x^{n-1}$:
$I = \int \frac{x^{n-1}}{x \cdot x^{n-1}(x^n+1)} \, dx = \int \frac{x^{n-1}}{x^n(x^n+1)} \, dx$
[Per the property of algebraic manipulation where $\frac{a}{b} = \frac{a \cdot c}{b \cdot c}$ for $c \neq 0$].Step 2: Substitution and Differential Transformation
Let $t = x^n$. Then, differentiating both sides with respect to $x$ yields:
$\frac{dt}{dx} = nx^{n-1} \implies dt = nx^{n-1} \, dx \implies x^{n-1} \, dx = \frac{dt}{n}$
Substituting these into the integral expression:
$I = \int \frac{1}{t(t+1)} \cdot \frac{dt}{n} = \frac{1}{n} \int \frac{1}{t(t+1)} \, dt$
Step 3: Decomposition via Partial Fractions
We decompose the integrand $\frac{1}{t(t+1)}$ into partial fractions:
$\frac{1}{t(t+1)} = \frac{A}{t} + \frac{B}{t+1}$
$1 = A(t+1) + Bt$
- Setting $t = 0$: $1 = A(1) \implies A = 1$
- Setting $t = -1$: $1 = B(-1) \implies B = -1$
Step 4: Integration
Substituting the constants $A$ and $B$ back into the integral:
$I = \frac{1}{n} \int \left( \frac{1}{t} - \frac{1}{t+1} \right) \, dt$
$I = \frac{1}{n} \left[ \ln|t| - \ln|t+1| \right] + C$
$I = \frac{1}{n} \ln \left| \frac{t}{t+1} \right| + C$
[Per the standard logarithmic integral formula $\int \frac{1}{u} du = \ln|u| + C$].Step 5: Final Back-Substitution
Substitute $t = x^n$ back into the expression:
$I = \frac{1}{n} \ln \left| \frac{x^n}{x^n+1} \right| + C$
Final Solution: $\frac{1}{n} \ln \left| \frac{x^n}{x^n+1} \right| + C$
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. $\frac{1}{x^4-1}$
- 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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