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Q4:
Integrate the functions in Exercises 1 to 22.
4. $x \log x$
Solution :
Step 1: Application of Integration by Parts
To evaluate the integral $I = \int x \log x \, dx$, we utilize the method of Integration by Parts, which is defined by the formula:
$\int u(x)v'(x) \, dx = u(x)v(x) - \int u'(x)v(x) \, dx$ [Per the Integration by Parts Rule]
We apply the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) to select our functions:
- Let $u = \log x$ (Logarithmic function)
- Let $dv = x \, dx$ (Algebraic function)
Step 2: Differentiation and Integration of Selected Components
Differentiating $u$ with respect to $x$ and integrating $dv$:
- $du = \frac{d}{dx}(\log x) \, dx = \frac{1}{x} \, dx$
- $v = \int x \, dx = \frac{x^2}{2}$ [By the Power Rule of Integration, $\int x^n \, dx = \frac{x^{n+1}}{n+1}$]
Step 3: Substitution into the Formula
Substituting these values into the Integration by Parts formula:
$I = (\log x) \cdot \left( \frac{x^2}{2} \right) - \int \left( \frac{x^2}{2} \right) \cdot \left( \frac{1}{x} \right) \, dx$
Step 4: Algebraic Simplification and Final Integration
Simplify the integrand within the integral expression:
$I = \frac{x^2}{2} \log x - \frac{1}{2} \int x \, dx$
Perform the final integration of $x$:
$I = \frac{x^2}{2} \log x - \frac{1}{2} \left( \frac{x^2}{2} \right) + C$
$I = \frac{x^2}{2} \log x - \frac{x^2}{4} + C$
Visual Representation of the Resulting Function
Step 5: Final Result
Factoring the expression for clarity:
$I = \frac{x^2}{4} (2 \log x - 1) + C$
Final Solution: $\frac{x^2}{2} \log x - \frac{x^2}{4} + C$ or $\frac{x^2}{4} (2 \log x - 1) + C$
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. $x^2 e^x$
- 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
- 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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