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Q12:
Integrate the functions in Exercises 1 to 22.
12. $x \sec^2 x$
Solution :
Initial Setup: Identifying the Method of Integration
To evaluate the integral $I = \int x \sec^2 x \, dx$, we apply the method of Integration by Parts. The formula for Integration by Parts, derived from the product rule of differentiation, is given by:
$\int u(x) v'(x) \, dx = u(x)v(x) - \int u'(x) v(x) \, dx$ [Per the Integration by Parts Theorem]
We select $u(x)$ and $v'(x)$ based on the ILATE rule (Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential), which dictates the hierarchy for choosing $u(x)$.
Step 1: Assigning Variables
- Let $u = x$ (Algebraic function). [Choosing $u$ to be the function that simplifies upon differentiation].
- Let $dv = \sec^2 x \, dx$ (Trigonometric function).
- Then, differentiate $u$: $du = dx$.
- Then, integrate $dv$: $v = \int \sec^2 x \, dx = \tan x$. [Using the standard integral formula $\int \sec^2 x \, dx = \tan x + C$].
Step 2: Applying the Integration by Parts Formula
Substitute the identified components into the integration formula:
$I = x \tan x - \int \tan x \, dx$
Step 3: Evaluating the Remaining Integral
The integral of $\tan x$ is a standard integral derived from the substitution $u = \cos x$:
$\int \tan x \, dx = \int \frac{\sin x}{\cos x} \, dx = -\ln|\cos x| + C = \ln|\sec x| + C$ [Using the property $\ln(a^{-1}) = -\ln(a)$].
Step 4: Final Assembly
Substitute the result of the integral back into the expression for $I$:
$I = x \tan x - (-\ln|\cos x|) + C$
$I = x \tan x + \ln|\cos x| + C$
Visual Representation of the Functions
Final Solution: $x \tan x + \ln|\sec x| + C$
More Questions from Class 12 Mathematics Integrals EXERCISE 7.6
- Q1: Integrate the functions in Exercises 1 to 22. 1. $x \sin x$
- Q2: Integrate the functions in Exercises 1 to 22. 2. $x \sin 3x$
- Q3: Integrate the functions in Exercises 1 to 22. 3. $x^2 e^x$
- Q4: Integrate the functions in Exercises 1 to 22. 4. $x \log x$
- Q5: Integrate the functions in Exercises 1 to 22. 5. $x \log 2x$
- Q6: Integrate the functions in Exercises 1 to 22. 6. $x^2 \log x$
- Q7: Integrate the functions in Exercises 1 to 22. 7. $x \sin^{-1} x$
- Q8: Integrate the functions in Exercises 1 to 22. 8. $x \tan^{-1} x$
- Q9: Integrate the functions in Exercises 1 to 22. 9. $x \cos^{-1} x$
- Q10: Integrate the functions in Exercises 1 to 22. 10. $(\sin^{-1} x)^2$
- Q11: Integrate the functions in Exercises 1 to 22. 11. $\frac{x \cos^{-1} x}{\sqrt{1-x^2}}$
- Q13: Integrate the functions in Exercises 1 to 22. 13. $\tan^{-1} x$
- Q14: Integrate the functions in Exercises 1 to 22. 14. $x (\log x)^2$
- Q15: Integrate the functions in Exercises 1 to 22. 15. $(x^2+1) \log x$
- Q16: Integrate the functions in Exercises 1 to 22. 16. $e^x(\sin x + \cos x)$
- Q17: Integrate the functions in Exercises 1 to 22. 17. $\frac{x e^x}{(1+x)^2}$
- Q18: Integrate the functions in Exercises 1 to 22. 18. $e^x (\frac{1+\sin x}{1+\cos x})$
- Q19: Integrate the functions in Exercises 1 to 22. 19. $e^x (\frac{1}{x} - \frac{1}{x^2})$
- Q20: Integrate the functions in Exercises 1 to 22. 20. $\frac{(x-3)e^x}{(x-1)^3}$
- Q21: Integrate the functions in Exercises 1 to 22. 21. $e^{2x} \sin x$
- Q22: Integrate the functions in Exercises 1 to 22. 22. $\sin^{-1} (\frac{2x}{1+x^2})$
- Q23: Choose the correct answer in Exercises 23 and 24. 23. $\int x^2 e^{x^3} dx$ equals
- Q24: Choose the correct answer in Exercises 23 and 24. 24. $\int e^x \sec x (1 + \tan x) dx$ 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 7.3
- Integrals Class 12 EXERCISE 7.4
- Integrals Class 12 EXERCISE 7.5
- 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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