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Q21:
Choose the correct answer in Exercises 21 and 22.
21. $\int_{1}^{\sqrt{3}} \frac{dx}{1+x^2}$ equals
Solution :
Step 1: Identification of the Integral Form
The given definite integral is:
$I = \int_{1}^{\sqrt{3}} \frac{dx}{1+x^2}$
We observe that the integrand is of the standard form $\frac{1}{a^2 + x^2}$, where $a = 1$. [Per the standard integral table for inverse trigonometric functions]
Step 2: Applying the Fundamental Theorem of Calculus
We recall the standard result for the antiderivative:
$\int \frac{dx}{1+x^2} = \tan^{-1}(x) + C$
Applying the Fundamental Theorem of Calculus, Part II, which states that if $F$ is the antiderivative of $f$, then $\int_{a}^{b} f(x) dx = F(b) - F(a)$:
$I = [\tan^{-1}(x)]_{1}^{\sqrt{3}}$
Step 3: Evaluation of the Definite Integral
Substitute the upper and lower limits into the evaluated antiderivative:
$I = \tan^{-1}(\sqrt{3}) - \tan^{-1}(1)$
Step 4: Calculation of Trigonometric Values
Using the principal values of the inverse tangent function:
- $\tan^{-1}(\sqrt{3}) = \frac{\pi}{3}$ [Since $\tan(\frac{\pi}{3}) = \sqrt{3}$]
- $\tan^{-1}(1) = \frac{\pi}{4}$ [Since $\tan(\frac{\pi}{4}) = 1$]
Therefore:
$I = \frac{\pi}{3} - \frac{\pi}{4}$
Step 5: Final Algebraic Simplification
To subtract the fractions, find the least common denominator (LCD), which is $12$:
$I = \frac{4\pi}{12} - \frac{3\pi}{12} = \frac{\pi}{12}$
Final Solution: \frac{\pi}{12}
More Questions from Class 12 Mathematics Integrals EXERCISE 7.8
- Q1: Evaluate the definite integrals in Exercises 1 to 20. 1. $\int_{-1}^{1} (x+1) dx$
- Q2: Evaluate the definite integrals in Exercises 1 to 20. 2. $\int_{2}^{3} \frac{1}{x} dx$
- Q3: Evaluate the definite integrals in Exercises 1 to 20. 3. $\int_{1}^{2} (4x^3 - 5x^2 + 6x + 9) dx$
- Q4: Evaluate the definite integrals in Exercises 1 to 20. 4. $\int_{0}^{\frac{\pi}{4}} \sin 2x dx$
- Q5: Evaluate the definite integrals in Exercises 1 to 20. 5. $\int_{0}^{\frac{\pi}{2}} \cos 2x dx$
- Q6: Evaluate the definite integrals in Exercises 1 to 20. 6. $\int_{4}^{5} e^x dx$
- Q7: Evaluate the definite integrals in Exercises 1 to 20. 7. $\int_{0}^{\frac{\pi}{4}} \tan x dx$
- Q8: Evaluate the definite integrals in Exercises 1 to 20. 8. $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \csc x dx$
- Q9: Evaluate the definite integrals in Exercises 1 to 20. 9. $\int_{0}^{1} \frac{dx}{\sqrt{1-x^2}}$
- Q10: Evaluate the definite integrals in Exercises 1 to 20. 10. $\int_{0}^{1} \frac{dx}{1+x^2}$
- Q11: Evaluate the definite integrals in Exercises 1 to 20. 11. $\int_{2}^{3} \frac{dx}{x^2-1}$
- Q12: Evaluate the definite integrals in Exercises 1 to 20. 12. $\int_{0}^{\frac{\pi}{2}} \cos^2 x dx$
- Q13: Evaluate the definite integrals in Exercises 1 to 20. 13. $\int_{2}^{3} \frac{x dx}{x^2+1}$
- Q14: Evaluate the definite integrals in Exercises 1 to 20. 14. $\int_{0}^{1} \frac{2x+3}{5x^2+1} dx$
- Q15: Evaluate the definite integrals in Exercises 1 to 20. 15. $\int_{0}^{1} x e^{x^2} dx$
- Q16: Evaluate the definite integrals in Exercises 1 to 20. 16. $\int_{1}^{2} \frac{5x^2}{x^2+4x+3} dx$
- Q17: Evaluate the definite integrals in Exercises 1 to 20. 17. $\int_{0}^{\frac{\pi}{4}} (2\sec^2 x + x^3 + 2) dx$
- Q18: Evaluate the definite integrals in Exercises 1 to 20. 18. $\int_{0}^{\pi} (\sin^2 \frac{x}{2} - \cos^2 \frac{x}{2}) dx$
- Q19: Evaluate the definite integrals in Exercises 1 to 20. 19. $\int_{0}^{2} \frac{6x+3}{x^2+4} dx$
- Q20: Evaluate the definite integrals in Exercises 1 to 20. 20. $\int_{0}^{1} (x e^x + \sin \frac{\pi x}{4}) dx$
- Q22: Choose the correct answer in Exercises 21 and 22. 22. $\int_{0}^{\frac{2}{3}} \frac{dx}{4+9x^2}$ 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 7.6
- Integrals Class 12 EXERCISE 7.7
- Integrals Class 12 EXERCISE 7.9
- Integrals Class 12 Miscellaneous Exercise on Chapter 7
Chapters in CBSE - Class 12 Mathematics
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