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Q17:
Integrate the rational functions in Exercises 1 to 21.
17. $\frac{\cos x}{(1-\sin x)(2-\sin x)}$ [Hint : Put $\sin x = t$]
Solution :
Step 1: Substitution and Variable Transformation
Given the integral: $I = \int \frac{\cos x}{(1-\sin x)(2-\sin x)} dx$.
Following the hint, let $\sin x = t$. Differentiating both sides with respect to $x$:
$\frac{d}{dx}(\sin x) = \frac{dt}{dx} \implies \cos x \, dx = dt$ [By the Chain Rule of differentiation].
Substituting these into the integral, the expression transforms into a rational function in terms of $t$:
$I = \int \frac{1}{(1-t)(2-t)} dt$
Step 2: Partial Fraction Decomposition
To integrate $\frac{1}{(1-t)(2-t)}$, we decompose the integrand into partial fractions:
$\frac{1}{(1-t)(2-t)} = \frac{A}{1-t} + \frac{B}{2-t}$
Multiplying both sides by the denominator $(1-t)(2-t)$:
$1 = A(2-t) + B(1-t)$
To solve for $A$ and $B$, we use the method of strategic substitution:
- Set $t = 1$: $1 = A(2-1) + B(0) \implies A = 1$.
- Set $t = 2$: $1 = A(2-2) + B(1-2) \implies 1 = -B \implies B = -1$.
[By the Identity Principle for polynomials].
Step 3: Integration of the Partial Fractions
Substituting $A=1$ and $B=-1$ back into the integral expression:
$I = \int \left( \frac{1}{1-t} - \frac{1}{2-t} \right) dt$
Applying the standard integral formula $\int \frac{1}{ax+b} dx = \frac{1}{a} \ln|ax+b| + C$:
$I = \int \frac{1}{1-t} dt - \int \frac{1}{2-t} dt$
$I = \frac{\ln|1-t|}{-1} - \frac{\ln|2-t|}{-1} + C$ [Applying the Linear Substitution Rule].
$I = -\ln|1-t| + \ln|2-t| + C$
Step 4: Final Substitution and Simplification
Using the logarithmic property $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$:
$I = \ln\left| \frac{2-t}{1-t} \right| + C$
Substitute $t = \sin x$ back into the result to obtain the function in terms of $x$:
$I = \ln\left| \frac{2-\sin x}{1-\sin x} \right| + C$
Geometric Representation of the Integral Mapping
Final Solution: \ln\left| \frac{2-\sin x}{1-\sin x} \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 $x^n = t$]
- 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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