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Q19:
Find the following integrals 19. $\int \frac{\sec^2 x}{\csc^2 x} dx$
Find the following integrals 19. $\int \frac{\sec^2 x}{\csc^2 x} dx$
Solution :
Step 1: Trigonometric Simplification
To evaluate the integral $\int \frac{\sec^2 x}{\csc^2 x} dx$, we must first express the integrand in terms of fundamental trigonometric functions, namely $\sin x$ and $\cos x$.
Using the reciprocal identities:
- $\sec x = \frac{1}{\cos x} \implies \sec^2 x = \frac{1}{\cos^2 x}$ [Reciprocal Identity]
- $\csc x = \frac{1}{\sin x} \implies \csc^2 x = \frac{1}{\sin^2 x}$ [Reciprocal Identity]
Substituting these into the integrand:
$\frac{\sec^2 x}{\csc^2 x} = \frac{\frac{1}{\cos^2 x}}{\frac{1}{\sin^2 x}} = \frac{1}{\cos^2 x} \cdot \frac{\sin^2 x}{1} = \frac{\sin^2 x}{\cos^2 x}$
Applying the quotient identity, $\frac{\sin x}{\cos x} = \tan x$, we obtain:
$\frac{\sin^2 x}{\cos^2 x} = \tan^2 x$ [Quotient Identity]
Step 2: Algebraic Manipulation for Integration
The standard integral table does not provide a direct primitive for $\tan^2 x$. We utilize the Pythagorean identity to express $\tan^2 x$ in a form that is directly integrable:
$1 + \tan^2 x = \sec^2 x \implies \tan^2 x = \sec^2 x - 1$ [Pythagorean Identity]
Thus, the integral becomes:
$I = \int (\sec^2 x - 1) dx$
Step 3: Execution of the Integration
By the Linearity Property of integrals, we split the expression into two separate integrals:
$I = \int \sec^2 x \, dx - \int 1 \, dx$
Applying the fundamental integration rules:
- $\int \sec^2 x \, dx = \tan x + C_1$ [Standard Integral Formula]
- $\int 1 \, dx = x + C_2$ [Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1}$ where $n=0$]
Combining these, where $C = C_1 - C_2$ is the constant of integration:
$I = \tan x - x + C$
Visual Representation of the Integrand Transformation
Final Solution: $\tan x - x + C$
More Questions from Class 12 Mathematics Integrals EXERCISE 7.1
- Q1: Find an anti derivative (or integral) of the following functions by the method of inspection. 1. $\sin 2x$
- Q2: Find an anti derivative (or integral) of the following functions by the method of inspection. 2. $\cos 3x$
- Q3: Find an anti derivative (or integral) of the following functions by the method of inspection. 3. $e^{2x}$
- Q4: Find an anti derivative (or integral) of the following functions by the method of inspection. 4. $(ax + b)^2$
- Q5: Find an anti derivative (or integral) of the following functions by the method of inspection. 5. $\sin 2x - 4e^{3x}$
- Q6: Find the following integrals in Exercises 6 to 20: 6. $\int (4e^{3x} + 1) dx$
- Q7: Find the following integrals in Exercises 6 to 20: 7. $\int x^2(1 - \frac{1}{x^2}) dx$
- Q8: Find the following integrals 8. $\int (ax^2 + bx + c) dx$
- Q9: Find the following integrals 9. $\int (2x^2 + e^x) dx$
- Q10: Find the following integrals 10. $\int (\sqrt{x} - \frac{1}{\sqrt{x}})^2 dx$
- Q11: Find the following integrals 11. $\int \frac{x^3 + 5x^2 - 4}{x^2} dx$
- Q12: Find the following integrals 12. $\int \frac{x^3 + 3x + 4}{\sqrt{x}} dx$
- Q13: Find the following integrals 13. $\int \frac{x^3 - x^2 + x - 1}{x - 1} dx$
- Q14: Find the following integrals 14. $\int (1 - x)\sqrt{x} dx$
- Q15: Find the following integrals 15. $\int \sqrt{x} (3x^2 + 2x + 3) dx$
- Q16: Find the following integrals 16. $\int (2x - 3\cos x + e^x) dx$
- Q17: Find the following integrals 17. $\int (2x^2 - 3\sin x + 5\sqrt{x}) dx$
- Q18: Find the following integrals 18. $\int \sec x (\sec x + \tan x) dx$
- Q20: Find the following integrals 20. $\int \frac{2 - 3\sin x}{\cos^2 x} dx$
- Q21: Choose the correct answer 21. The anti derivative of $(\sqrt{x} + \frac{1}{\sqrt{x}})$ equals
- Q22: Choose the correct answer 22. If $\frac{d}{dx} f(x) = 4x^3 - \frac{3}{x^4}$ such that $f(2) = 0$. Then $f(x)$ is
CBSE Solutions for Class 12 Mathematics Integrals
- Integrals Class 12 EXERCISE 7.10
- Integrals Class 12 EXERCISE 7.2
- Integrals Class 12 EXERCISE 7.3
- 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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