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Q2:
Find the integrals of the functions in Exercises 1 to 22:
2. $\sin 3x \cos 4x$
Solution :
Step 1: Application of Trigonometric Product-to-Sum Identities
To evaluate the integral $\int \sin 3x \cos 4x \, dx$, we must transform the product of trigonometric functions into a sum or difference. We utilize the standard trigonometric identity [Product-to-Sum Formula]:
$\sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)]$
Setting $A = 3x$ and $B = 4x$, we substitute these values into the identity:
$\sin 3x \cos 4x = \frac{1}{2} [\sin(3x + 4x) + \sin(3x - 4x)]$
$\sin 3x \cos 4x = \frac{1}{2} [\sin 7x + \sin(-x)]$
Using the parity property of the sine function [$\sin(-x) = -\sin x$], the expression simplifies to:
$\sin 3x \cos 4x = \frac{1}{2} [\sin 7x - \sin x]$
Step 2: Integration of the Transformed Expression
Substitute the transformed expression back into the integral:
$I = \int \frac{1}{2} (\sin 7x - \sin x) \, dx$
$I = \frac{1}{2} \int \sin 7x \, dx - \frac{1}{2} \int \sin x \, dx$
Step 3: Execution of Integration
Applying the standard integral formula [$\int \sin(ax) \, dx = -\frac{\cos(ax)}{a} + C$]:
For the first term: $\int \sin 7x \, dx = -\frac{\cos 7x}{7}$
For the second term: $\int \sin x \, dx = -\cos x$
Combining these into the expression:
$I = \frac{1}{2} \left[ -\frac{\cos 7x}{7} - (-\cos x) \right] + C$
$I = \frac{1}{2} \left[ \cos x - \frac{\cos 7x}{7} \right] + C$
$I = \frac{\cos x}{2} - \frac{\cos 7x}{14} + C$
Visual Representation of the Integrand
Final Solution: $\frac{\cos x}{2} - \frac{\cos 7x}{14} + C$
More Questions from Class 12 Mathematics Integrals EXERCISE 7.3
- Q1: Find the integrals of the functions in Exercises 1 to 22: 1. $\sin^2(2x+5)$
- 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. $\tan^4 x$
- 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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