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Q7:
Find the integrals of the functions in Exercises 1 to 22:
7. $\sin 4x \sin 8x$
Solution :
Step 1: Application of Trigonometric Identity
To integrate the product of two sine functions, we utilize the product-to-sum trigonometric identity: $\sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)]$ [By Product-to-Sum Identity].
Given the integral: $I = \int \sin(4x) \sin(8x) \, dx$
Let $A = 8x$ and $B = 4x$ (choosing the larger argument first simplifies the sign): $I = \int \frac{1}{2} [\cos(8x - 4x) - \cos(8x + 4x)] \, dx$
Step 2: Simplification of the Integrand
Substituting the values into the integral: $I = \frac{1}{2} \int [\cos(4x) - \cos(12x)] \, dx$
Step 3: Execution of Term-by-Term Integration
Applying the linearity property of integrals [$\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx$]: $I = \frac{1}{2} \left[ \int \cos(4x) \, dx - \int \cos(12x) \, dx \right]$
Using the standard integral formula $\int \cos(kx) \, dx = \frac{\sin(kx)}{k} + C$:
- For $\int \cos(4x) \, dx$, we get $\frac{\sin(4x)}{4}$
- For $\int \cos(12x) \, dx$, we get $\frac{\sin(12x)}{12}$
Step 4: Algebraic Consolidation
Combining the results: $I = \frac{1}{2} \left[ \frac{\sin(4x)}{4} - \frac{\sin(12x)}{12} \right] + C$
Distributing the scalar $\frac{1}{2}$: $I = \frac{\sin(4x)}{8} - \frac{\sin(12x)}{24} + C$
Visual Representation of Integral Decomposition
Final Solution: $\frac{\sin(4x)}{8} - \frac{\sin(12x)}{24} + C$
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. $\sin x \sin 2x \sin 3x$
- 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. $\tan^4 x$
- Q17: Find the integrals of the functions in Exercises 1 to 22: 17. $\frac{\sin^3 x + \cos^3 x}{\sin^2 x \cos^2 x}$
- 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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