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Q18:
Find the integrals of the functions in Exercises 1 to 22:
18. $\frac{\cos 2x + 2\sin^2 x}{\cos^2 x}$
Solution :
Step 1: Application of Trigonometric Identities
The integrand is given by:
$I = \int \frac{\cos 2x + 2\sin^2 x}{\cos^2 x} \, dx$To simplify the numerator, apply the double-angle identity for cosine: $\cos 2x = 1 - 2\sin^2 x$ [Per the trigonometric identity $\cos 2\theta = 1 - 2\sin^2 \theta$].
Substituting this into the integral expression:
$I = \int \frac{(1 - 2\sin^2 x) + 2\sin^2 x}{\cos^2 x} \, dx$Step 2: Algebraic Simplification
Observe the cancellation of the $2\sin^2 x$ terms in the numerator:
$I = \int \frac{1}{\cos^2 x} \, dx$Utilizing the reciprocal trigonometric identity, $\frac{1}{\cos^2 x} = \sec^2 x$ [By definition of the secant function], the integral simplifies to:
$I = \int \sec^2 x \, dx$Step 3: Integration and Final Result
Applying the standard integral formula for the square of the secant function [Per the Fundamental Theorem of Calculus/Table of Integrals]:
$\int \sec^2 x \, dx = \tan x + C$where $C$ represents the constant of integration.
Visual Representation of the Integrand Function
Final Solution: $\tan x + C$
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. $\tan^4 x$
- Q17: Find the integrals of the functions in Exercises 1 to 22: 17. $\frac{\sin^3 x + \cos^3 x}{\sin^2 x \cos^2 x}$
- 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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