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Q22:
Integrate the functions in Exercises 1 to 22.
22. $\sin^{-1} (\frac{2x}{1+x^2})$
Solution :
Step 1: Trigonometric Substitution
Given the integrand $I = \int \sin^{-1} \left( \frac{2x}{1+x^2} \right) dx$. We observe the form of the argument within the inverse trigonometric function resembles the double-angle formula for sine, specifically $\sin(2\theta) = \frac{2\tan\theta}{1+\tan^2\theta}$.
Let $x = \tan\theta$. Then, $dx = \sec^2\theta \, d\theta$ [By the method of substitution].
The integrand transforms as follows:
$\sin^{-1} \left( \frac{2\tan\theta}{1+\tan^2\theta} \right) = \sin^{-1}(\sin 2\theta) = 2\theta$ [For $\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$, which implies $x \in [-1, 1]$].
Step 2: Constructing the Integral
Substituting the differential and the transformed function into the integral:
$I = \int (2\theta) \cdot \sec^2\theta \, d\theta = 2 \int \theta \sec^2\theta \, d\theta$
Step 3: Integration by Parts
We utilize the formula for integration by parts: $\int u \, dv = uv - \int v \, du$.
Let $u = \theta$ and $dv = \sec^2\theta \, d\theta$.
Then, $du = d\theta$ and $v = \int \sec^2\theta \, d\theta = \tan\theta$ [By basic integral of secant squared].
$I = 2 \left[ \theta \tan\theta - \int \tan\theta \, d\theta \right]$
Step 4: Evaluating the Resulting Integral
The integral of $\tan\theta$ is $\ln|\sec\theta|$ or $-\ln|\cos\theta|$.
$I = 2 \left[ \theta \tan\theta - \ln|\sec\theta| \right] + C$
Since $\sec\theta = \sqrt{1+\tan^2\theta} = \sqrt{1+x^2}$, we can substitute back in terms of $x$.
Step 5: Back-Substitution
We have $\theta = \tan^{-1} x$, $\tan\theta = x$, and $\sec\theta = \sqrt{1+x^2}$.
$I = 2 [x \tan^{-1} x - \ln|\sqrt{1+x^2}|] + C$
Using the logarithm property $\ln(a^b) = b \ln a$:
$I = 2x \tan^{-1} x - 2 \cdot \frac{1}{2} \ln(1+x^2) + C$
Final Solution: $2x \tan^{-1} x - \ln(1+x^2) + C$
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. $e^x (\frac{1+\sin x}{1+\cos x})$
- 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. $e^{2x} \sin x$
- 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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