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Q19:
Integrate the rational functions in Exercises 1 to 21.
19. $\frac{2x}{(x^2+1)(x^2+3)}$
Solution :
Step 1: Substitution for Simplification
To integrate the function $I = \int \frac{2x}{(x^2+1)(x^2+3)} dx$, we observe that the derivative of $x^2$ is $2x$. Let us use the method of substitution.
Let $t = x^2$. [By the Method of Substitution]
Then, the differential $dt = 2x \, dx$.
The integral transforms into:
$I = \int \frac{1}{(t+1)(t+3)} dt$Step 2: Partial Fraction Decomposition
We express the integrand as a sum of partial fractions:
$\frac{1}{(t+1)(t+3)} = \frac{A}{t+1} + \frac{B}{t+3}$Multiplying both sides by $(t+1)(t+3)$, we obtain the identity:
$1 = A(t+3) + B(t+1)$To find the constants $A$ and $B$, we use the method of equating coefficients or substituting critical values: [By Partial Fraction Theorem]
- Set $t = -1$: $1 = A(-1+3) \implies 1 = 2A \implies A = \frac{1}{2}$
- Set $t = -3$: $1 = B(-3+1) \implies 1 = -2B \implies B = -\frac{1}{2}$
Step 3: Evaluating the Integral
Substitute the constants back into the integral form:
$I = \int \left( \frac{1/2}{t+1} - \frac{1/2}{t+3} \right) dt$ $I = \frac{1}{2} \int \frac{1}{t+1} dt - \frac{1}{2} \int \frac{1}{t+3} dt$Using the standard integral $\int \frac{1}{x} dx = \ln|x| + C$:
$I = \frac{1}{2} \ln|t+1| - \frac{1}{2} \ln|t+3| + C$ $I = \frac{1}{2} \ln\left| \frac{t+1}{t+3} \right| + C$Step 4: Re-substitution and Simplification
Substituting $t = x^2$ back into the result:
$I = \frac{1}{2} \ln\left| \frac{x^2+1}{x^2+3} \right| + C$Since $x^2+1 > 0$ and $x^2+3 > 0$ for all real $x$, the absolute value bars are redundant.
Final Solution: \frac{1}{2} \ln\left( \frac{x^2+1}{x^2+3} \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. $\frac{(x^2+1)(x^2+2)}{(x^2+3)(x^2+4)}$
- 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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