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Q5:
Integrate the functions in Exercises 1 to 22.
5. $x \log 2x$
Solution :
Step 1: Application of Integration by Parts Formula
To evaluate the integral $\int x \log(2x) \, dx$, we utilize the method of Integration by Parts, which states: $\int u \cdot v \, dx = u \int v \, dx - \int \left( \frac{du}{dx} \cdot \int v \, dx \right) dx$ [Integration by Parts Formula]
According to the ILATE rule (Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential), we assign the functions as follows:
- Let $u = \log(2x)$ [Logarithmic function chosen as the first function]
- Let $dv = x \, dx$ [Algebraic function chosen as the second function]
Step 2: Differentiation and Integration of Assigned Parts
Calculate the derivative of $u$ and the integral of $dv$:
- $\frac{du}{dx} = \frac{d}{dx}(\log(2x)) = \frac{1}{2x} \cdot 2 = \frac{1}{x}$ [Chain Rule of Differentiation]
- $v = \int x \, dx = \frac{x^2}{2}$ [Power Rule of Integration: $\int x^n dx = \frac{x^{n+1}}{n+1}$]
Step 3: Substitution into the Formula
Substitute these components into the Integration by Parts formula: $\int x \log(2x) \, dx = \left( \log(2x) \cdot \frac{x^2}{2} \right) - \int \left( \frac{1}{x} \cdot \frac{x^2}{2} \right) dx$
Step 4: Algebraic Simplification and Final Integration
Simplify the integrand within the integral sign: $\int x \log(2x) \, dx = \frac{x^2}{2} \log(2x) - \int \frac{x}{2} \, dx$ $\int x \log(2x) \, dx = \frac{x^2}{2} \log(2x) - \frac{1}{2} \int x \, dx$
Perform the final integration: $= \frac{x^2}{2} \log(2x) - \frac{1}{2} \left( \frac{x^2}{2} \right) + C$ [Where $C$ is the constant of integration]
Step 5: Final Expression Consolidation
Combining the terms: $= \frac{x^2}{2} \log(2x) - \frac{x^2}{4} + C$
Final Solution: $\frac{x^2}{2} \log(2x) - \frac{x^2}{4} + C$
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. $x \log x$
- 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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