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Q21:
Find the integrals of the functions in Exercises 1 to 22:
21. $\sin^{-1}(\cos x)$
Solution :
Step 1: Simplifying the Integrand
The given integral is $I = \int \sin^{-1}(\cos x) \, dx$. We utilize the trigonometric identity $\cos x = \sin\left(\frac{\pi}{2} - x\right)$ to resolve the inverse trigonometric function. [By the property of complementary angles in trigonometry].
Substituting this identity into the expression:
$I = \int \sin^{-1}\left(\sin\left(\frac{\pi}{2} - x\right)\right) \, dx$
Since the principal value branch of $\sin^{-1}(x)$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, the expression simplifies as follows [Per the definition of the composition of inverse trigonometric functions]:
$I = \int \left(\frac{\pi}{2} - x\right) \, dx$
Step 2: Integration via Linearity Property
The integral of a sum/difference is the sum/difference of the integrals. Applying the linearity property of indefinite integrals [Per the fundamental theorem of calculus]:
$I = \int \frac{\pi}{2} \, dx - \int x \, dx$
Step 3: Executing the Integration
We apply the constant integration rule $\int a \, dx = ax + C$ and the power rule for integration $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$:
- Integral of constant $\frac{\pi}{2}$ is $\frac{\pi}{2}x$.
- Integral of $x^1$ is $\frac{x^{1+1}}{1+1} = \frac{x^2}{2}$.
$I = \frac{\pi}{2}x - \frac{x^2}{2} + C$
Visual Representation of the Linear Simplification
Step 4: Final Consolidation
Combining the terms obtained in Step 3, we arrive at the final antiderivative. $C$ represents the constant of integration, which is required for all indefinite integrals [By the definition of the antiderivative family].
Final Solution: $\frac{\pi x}{2} - \frac{x^2}{2} + C$
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. $\frac{\sin^3 x + \cos^3 x}{\sin^2 x \cos^2 x}$
- 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. $\frac{\cos 2x}{(\cos x + \sin x)^2}$
- 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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