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Q2:
Integrate the functions in Exercises 1 to 22.
2. $x \sin 3x$
Solution :
Step 1: Identifying the Integration Technique
To evaluate the integral $I = \int x \sin 3x \, dx$, we employ the method of Integration by Parts. This method is defined by the formula:
$\int u(x) v(x) \, dx = u(x) \int v(x) \, dx - \int \left( \frac{du}{dx} \int v(x) \, dx \right) \, dx$ [Per the Integration by Parts Rule]
Step 2: Selecting Functions (ILATE Rule)
Following the ILATE hierarchy (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential), we assign the functions:
- $u = x$ (Algebraic function)
- $dv = \sin 3x \, dx$ (Trigonometric function)
Calculating the derivatives and integrals:
- $\frac{du}{dx} = 1 \implies du = dx$
- $v = \int \sin 3x \, dx = -\frac{\cos 3x}{3}$ [Per the integration rule $\int \sin(ax) \, dx = -\frac{\cos(ax)}{a} + C$]
Step 3: Executing the Integration by Parts Formula
Substitute $u, v, du,$ and $dv$ into the formula:
$I = x \left( -\frac{\cos 3x}{3} \right) - \int \left( 1 \cdot -\frac{\cos 3x}{3} \right) \, dx$
$I = -\frac{x \cos 3x}{3} + \frac{1}{3} \int \cos 3x \, dx$
Step 4: Final Evaluation of the Integral
Integrate $\cos 3x$:
$\int \cos 3x \, dx = \frac{\sin 3x}{3} + C$ [Per the integration rule $\int \cos(ax) \, dx = \frac{\sin(ax)}{a} + C$]
Substituting this back into the expression for $I$:
$I = -\frac{x \cos 3x}{3} + \frac{1}{3} \left( \frac{\sin 3x}{3} \right) + C$
$I = -\frac{x \cos 3x}{3} + \frac{\sin 3x}{9} + C$
Visual Representation of the Integrand Area
Final Solution: $-\frac{x \cos 3x}{3} + \frac{\sin 3x}{9} + C$
More Questions from Class 12 Mathematics Integrals EXERCISE 7.6
- Q1: Integrate the functions in Exercises 1 to 22. 1. $x \sin x$
- 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
- 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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