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Q24:
Choose the correct answer in Exercises 23 and 24.
24. $\int e^x \sec x (1 + \tan x) dx$ equals
Solution :
Step 1: Analyzing the Integral Structure
The given integral is:
$I = \int e^x \sec x (1 + \tan x) dx$
Distribute the $\sec x$ term within the parenthesis:
$I = \int e^x (\sec x + \sec x \tan x) dx$
Step 2: Identifying the Relevant Integral Identity
We utilize the standard integration identity involving exponential functions:
$\int e^x [f(x) + f'(x)] dx = e^x f(x) + C$
[Per the Fundamental Theorem of Calculus and the rule for integrating exponential functions of the form $e^x(f(x) + f'(x))$].
Step 3: Verification of Functions
Let $f(x) = \sec x$.
To verify the identity, compute the derivative $f'(x)$:
$f'(x) = \frac{d}{dx}(\sec x)$
$f'(x) = \sec x \tan x$ [By the derivative rule for trigonometric functions].
Since the integrand matches the form $e^x [f(x) + f'(x)]$, where $f(x) = \sec x$ and $f'(x) = \sec x \tan x$, the conditions of the identity are satisfied.
Step 4: Applying the Integral Formula
Substitute $f(x) = \sec x$ into the result of the identity $e^x f(x) + C$:
$I = e^x \sec x + C$
Visual Representation of the Exponential Growth and Function
Conclusion
Comparing the result $e^x \sec x + C$ with the standard options provided in the examination context, this solution corresponds precisely to the evaluated integral.
Final Solution: $e^x \sec x + C$
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. $\frac{x \cos^{-1} x}{\sqrt{1-x^2}}$
- Q12: Integrate the functions in Exercises 1 to 22. 12. $x \sec^2 x$
- 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
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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