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Q17:
Find the integrals of the functions in Exercises 1 to 22:
17. $\frac{\sin^3 x + \cos^3 x}{\sin^2 x \cos^2 x}$
Solution :
Step 1: Decomposition of the Integrand
Given the integral: $I = \int \frac{\sin^3 x + \cos^3 x}{\sin^2 x \cos^2 x} \, dx$
We apply the property of fractions $\frac{a+b}{c} = \frac{a}{c} + \frac{b}{c}$ to separate the terms:
$I = \int \left( \frac{\sin^3 x}{\sin^2 x \cos^2 x} + \frac{\cos^3 x}{\sin^2 x \cos^2 x} \right) \, dx$
[By the Additive Property of Integration]
Step 2: Algebraic Simplification
Simplify each fraction by canceling common trigonometric powers:
$\frac{\sin^3 x}{\sin^2 x \cos^2 x} = \frac{\sin x}{\cos^2 x} = \frac{\sin x}{\cos x} \cdot \frac{1}{\cos x} = \tan x \sec x$
$\frac{\cos^3 x}{\sin^2 x \cos^2 x} = \frac{\cos x}{\sin^2 x} = \frac{\cos x}{\sin x} \cdot \frac{1}{\sin x} = \cot x \csc x$
Substituting these into the integral expression:
$I = \int (\tan x \sec x + \cot x \csc x) \, dx$
Step 3: Integration using Standard Forms
Distribute the integral operator across the sum:
$I = \int \tan x \sec x \, dx + \int \cot x \csc x \, dx$
Applying the standard integration formulas:
- $\int \sec x \tan x \, dx = \sec x + C_1$ [Standard Derivative Inverse]
- $\int \csc x \cot x \, dx = -\csc x + C_2$ [Standard Derivative Inverse]
Step 4: Final Compilation
Combining the results into a single expression with the constant of integration $C = C_1 + C_2$:
$I = \sec x - \csc x + C$
Visual Representation of Trig Functions
Final Solution: \int \frac{\sin^3 x + \cos^3 x}{\sin^2 x \cos^2 x} \, dx = \sec x - \csc x + C
More Questions from Class 12 Mathematics Integrals EXERCISE 7.3
- Q1: Find the integrals of the functions in Exercises 1 to 22: 1. $\sin^2(2x+5)$
- Q2: Find the integrals of the functions in Exercises 1 to 22: 2. $\sin 3x \cos 4x$
- Q3: Find the integrals of the functions in Exercises 1 to 22: 3. $\cos 2x \cos 4x \cos 6x$
- Q4: Find the integrals of the functions in Exercises 1 to 22: 4. $\sin^3(2x+1)$
- Q5: Find the integrals of the functions in Exercises 1 to 22: 5. $\sin^3 x \cos^3 x$
- Q6: Find the integrals of the functions in Exercises 1 to 22: 6. $\sin x \sin 2x \sin 3x$
- Q7: Find the integrals of the functions in Exercises 1 to 22: 7. $\sin 4x \sin 8x$
- Q8: Find the integrals of the functions in Exercises 1 to 22: 8. $\frac{1-\cos x}{1+\cos x}$
- Q9: Find the integrals of the functions in Exercises 1 to 22: 9. $\frac{\cos x}{1+\cos x}$
- Q10: Find the integrals of the functions in Exercises 1 to 22: 10. $\sin^4 x$
- Q11: Find the integrals of the functions in Exercises 1 to 22: 11. $\cos^4 2x$
- Q12: Find the integrals of the functions in Exercises 1 to 22: 12. $\frac{\sin^2 x}{1+\cos x}$
- Q13: Find the integrals of the functions in Exercises 1 to 22: 13. $\frac{\cos 2x - \cos 2\alpha}{\cos x - \cos \alpha}$
- Q14: Find the integrals of the functions in Exercises 1 to 22: 14. $\frac{\cos x - \sin x}{1+\sin 2x}$
- Q15: Find the integrals of the functions in Exercises 1 to 22: 15. $\tan^3 2x \sec 2x$
- Q16: Find the integrals of the functions in Exercises 1 to 22: 16. $\tan^4 x$
- Q18: Find the integrals of the functions in Exercises 1 to 22: 18. $\frac{\cos 2x + 2\sin^2 x}{\cos^2 x}$
- Q19: Find the integrals of the functions in Exercises 1 to 22: 19. $\frac{1}{\sin x \cos^3 x}$
- Q20: Find the integrals of the functions in Exercises 1 to 22: 20. $\frac{\cos 2x}{(\cos x + \sin x)^2}$
- Q21: Find the integrals of the functions in Exercises 1 to 22: 21. $\sin^{-1}(\cos x)$
- Q22: Find the integrals of the functions in Exercises 1 to 22: 22. $\frac{1}{\cos(x-a)\cos(x-b)}$
- Q23: Choose the correct answer in Exercises 23 and 24. 23. $\int \frac{\sin^2 x - \cos^2 x}{\sin^2 x \cos^2 x} dx$ is equal to
- Q24: Choose the correct answer in Exercises 23 and 24. 24. $\int \frac{e^x(1+x)}{\cos^2(e^x x)} dx$ 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 7.4
- Integrals Class 12 EXERCISE 7.5
- 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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