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Q19:
Integrate the functions in Exercises 1 to 22.
19. $e^x (\frac{1}{x} - \frac{1}{x^2})$
Solution :
Given Integral
We are required to evaluate the indefinite integral:
$I = \int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx$
Step 1: Identifying the Mathematical Form
The integrand is of the form $e^x [f(x) + f'(x)]$. We utilize the standard integration identity derived from the product rule of differentiation [By the Theorem of Integration by Parts]:
$\int e^x [f(x) + f'(x)] dx = e^x f(x) + C$
Step 2: Defining the Functions
Let $f(x) = \frac{1}{x}$.
To verify the existence of the derivative component, we differentiate $f(x)$ with respect to $x$ using the Power Rule $\frac{d}{dx}(x^n) = nx^{n-1}$:
$f'(x) = \frac{d}{dx}(x^{-1}) = -1 \cdot x^{-2} = -\frac{1}{x^2}$
Since the derivative of $\frac{1}{x}$ is exactly $-\frac{1}{x^2}$, our integrand fits the required form perfectly.
Step 3: Application of the Integration Identity
Substituting $f(x) = \frac{1}{x}$ and $f'(x) = -\frac{1}{x^2}$ into the identity $\int e^x [f(x) + f'(x)] dx = e^x f(x) + C$, we obtain:
$I = \int e^x \left( \frac{1}{x} + \left( -\frac{1}{x^2} \right) \right) dx$
$I = e^x \cdot \left( \frac{1}{x} \right) + C$
Visual Representation of Function Relationship
Step 4: Final Verification
Differentiating the result to ensure precision [By the Fundamental Theorem of Calculus]:
$\frac{d}{dx} \left( \frac{e^x}{x} + C \right) = \frac{x \cdot \frac{d}{dx}(e^x) - e^x \cdot \frac{d}{dx}(x)}{x^2} + 0$ [Using the Quotient Rule $\frac{d}{dx}(\frac{u}{v}) = \frac{vu' - uv'}{v^2}$]
$= \frac{x e^x - e^x}{x^2} = \frac{e^x(x - 1)}{x^2} = e^x \left( \frac{x}{x^2} - \frac{1}{x^2} \right) = e^x \left( \frac{1}{x} - \frac{1}{x^2} \right)$
The derivative confirms our result is accurate.
Final Solution: $\frac{e^x}{x} + C$
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. $e^x (\frac{1+\sin x}{1+\cos x})$
- 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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