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Q9:
Integrate the functions in Exercises 1 to 22.
9. $x \cos^{-1} x$
Solution :
Step 1: Application of Integration by Parts
To evaluate the integral $I = \int x \cos^{-1} x \, dx$, we employ the method of Integration by Parts, which states: $\int u \cdot v \, dx = u \int v \, dx - \int \left( \frac{du}{dx} \cdot \int v \, dx \right) dx$ [Per the Integration by Parts Theorem].
According to the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential), we assign:
- $u = \cos^{-1} x$
- $dv = x \, dx$
Differentiating $u$ and integrating $dv$ yields:
- $\frac{du}{dx} = -\frac{1}{\sqrt{1-x^2}}$
- $v = \int x \, dx = \frac{x^2}{2}$
Step 2: Substitution into the Formula
Applying the components to the integration formula:
$I = \left( \cos^{-1} x \right) \left( \frac{x^2}{2} \right) - \int \left( -\frac{1}{\sqrt{1-x^2}} \right) \left( \frac{x^2}{2} \right) dx$
$I = \frac{x^2}{2} \cos^{-1} x + \frac{1}{2} \int \frac{x^2}{\sqrt{1-x^2}} \, dx$
Step 3: Evaluating the Remaining Integral
To integrate $\int \frac{x^2}{\sqrt{1-x^2}} \, dx$, we manipulate the numerator to create a term that matches the denominator:
$\int \frac{x^2}{\sqrt{1-x^2}} \, dx = \int \frac{-(1-x^2) + 1}{\sqrt{1-x^2}} \, dx$
$= -\int \frac{1-x^2}{\sqrt{1-x^2}} \, dx + \int \frac{1}{\sqrt{1-x^2}} \, dx$
$= -\int \sqrt{1-x^2} \, dx + \int \frac{1}{\sqrt{1-x^2}} \, dx$
Step 4: Applying Standard Integral Formulas
We use the following standard results:
- $\int \sqrt{a^2-x^2} \, dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right)$
- $\int \frac{1}{\sqrt{a^2-x^2}} \, dx = \sin^{-1}\left(\frac{x}{a}\right)$
Thus, $\int \frac{x^2}{\sqrt{1-x^2}} \, dx = -\left( \frac{x}{2}\sqrt{1-x^2} + \frac{1}{2}\sin^{-1} x \right) + \sin^{-1} x$
$= -\frac{x}{2}\sqrt{1-x^2} - \frac{1}{2}\sin^{-1} x + \sin^{-1} x = -\frac{x}{2}\sqrt{1-x^2} + \frac{1}{2}\sin^{-1} x$
Step 5: Final Assembly
Substitute the result back into the expression for $I$:
$I = \frac{x^2}{2} \cos^{-1} x + \frac{1}{2} \left[ -\frac{x}{2}\sqrt{1-x^2} + \frac{1}{2}\sin^{-1} x \right] + C$
$I = \frac{x^2}{2} \cos^{-1} x - \frac{x}{4}\sqrt{1-x^2} + \frac{1}{4}\sin^{-1} x + C$
Final Solution: $\frac{x^2}{2} \cos^{-1} x - \frac{x}{4}\sqrt{1-x^2} + \frac{1}{4}\sin^{-1} x + C$
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. $x \tan^{-1} x$
- 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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