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Q9:
Evaluate the definite integrals in Exercises 1 to 20.
9. $\int_{0}^{1} \frac{dx}{\sqrt{1-x^2}}$
Solution :
Step 1: Identification of the Integral Form
The given definite integral is:
$I = \int_{0}^{1} \frac{dx}{\sqrt{1-x^2}}$
This integral corresponds to the standard derivative form of the inverse trigonometric function $\arcsin(x)$. [Per the Fundamental Theorem of Calculus and standard integration tables, $\int \frac{1}{\sqrt{a^2-x^2}} dx = \sin^{-1}\left(\frac{x}{a}\right) + C$.]
Step 2: Determining the Antiderivative
In this specific case, $a = 1$. Therefore, the antiderivative of $f(x) = \frac{1}{\sqrt{1-x^2}}$ is:
$F(x) = \sin^{-1}(x)$
By the Second Fundamental Theorem of Calculus, if $f$ is continuous on $[a, b]$ and $F$ is an antiderivative of $f$, then:
$\int_{a}^{b} f(x) dx = F(b) - F(a)$
Step 3: Evaluating the Definite Integral
Substituting the limits $0$ and $1$ into the antiderivative $F(x) = \sin^{-1}(x)$:
$I = [\sin^{-1}(x)]_{0}^{1}$
$I = \sin^{-1}(1) - \sin^{-1}(0)$
Step 4: Trigonometric Evaluation
We evaluate the inverse sine values based on the unit circle definitions:
- $\sin^{-1}(1) = \frac{\pi}{2}$ [Since $\sin(\frac{\pi}{2}) = 1$ and $\frac{\pi}{2} \in [-\frac{\pi}{2}, \frac{\pi}{2}]$]
- $\sin^{-1}(0) = 0$ [Since $\sin(0) = 0$ and $0 \in [-\frac{\pi}{2}, \frac{\pi}{2}]$]
$I = \frac{\pi}{2} - 0 = \frac{\pi}{2}$
Visual Representation of the Function
Final Solution: \frac{\pi}{2}
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. $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \csc x dx$
- 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. $\int_{0}^{1} (x e^x + \sin \frac{\pi x}{4}) dx$
- Q21: Choose the correct answer in Exercises 21 and 22. 21. $\int_{1}^{\sqrt{3}} \frac{dx}{1+x^2}$ equals
- 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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