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Q2:
By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19.
2. $\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx$
Solution :
Step 1: Formal Definition of the Integral
Let the given definite integral be defined as $I$:
$I = \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx$ --- (Equation 1)
Step 2: Application of Definite Integral Property
We utilize the King's Property, which states that for any continuous function $f(x)$ on $[a, b]$:
$\int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a + b - x) \, dx$ [Ref: Property of Definite Integrals P4]
Here, $a = 0$ and $b = \frac{\pi}{2}$. Thus, we substitute $x$ with $(\frac{\pi}{2} - x)$:
$I = \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin(\frac{\pi}{2} - x)}}{\sqrt{\sin(\frac{\pi}{2} - x)} + \sqrt{\cos(\frac{\pi}{2} - x)}} \, dx$
Step 3: Simplification using Trigonometric Identities
Employing the co-function identities $\sin(\frac{\pi}{2} - x) = \cos x$ and $\cos(\frac{\pi}{2} - x) = \sin x$ [Ref: Trigonometric Complementary Angle Theorem]:
$I = \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} \, dx$ --- (Equation 2)
Step 4: Algebraic Summation of Integrals
Add Equation 1 and Equation 2:
$I + I = \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx + \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} \, dx$
$2I = \int_{0}^{\frac{\pi}{2}} \left( \frac{\sqrt{\sin x} + \sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} \right) \, dx$
$2I = \int_{0}^{\frac{\pi}{2}} 1 \, dx$
Step 5: Final Integration and Evaluation
Evaluating the integral of the constant $1$ with respect to $x$:
$2I = [x]_{0}^{\frac{\pi}{2}}$
$2I = \frac{\pi}{2} - 0$
$2I = \frac{\pi}{2}$
$I = \frac{\pi}{4}$
Geometric Representation (Integrand Behavior)
Final Solution: \frac{\pi}{4}
More Questions from Class 12 Mathematics Integrals EXERCISE 7.10
- Q1: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 1. $\int_{0}^{\frac{\pi}{2}} \cos^2 x dx$
- Q3: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 3. $\int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{3}{2}} x dx}{\sin^{\frac{3}{2}} x + \cos^{\frac{3}{2}} x}$
- Q4: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 4. $\int_{0}^{\frac{\pi}{2}} \frac{\cos^5 x dx}{\sin^5 x + \cos^5 x}$
- Q5: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 5. $\int_{-5}^{5} |x+2| dx$
- Q6: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 6. $\int_{2}^{8} |x-5| dx$
- Q7: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 7. $\int_{0}^{1} x(1-x)^n dx$
- Q8: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 8. $\int_{0}^{\frac{\pi}{4}} \log(1+\tan x) dx$
- Q9: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 9. $\int_{0}^{2} x \sqrt{2-x} dx$
- Q10: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 10. $\int_{0}^{\frac{\pi}{2}} (2\log \sin x - \log \sin 2x) dx$
- Q11: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 11. $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^2 x dx$
- Q12: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 12. $\int_{0}^{\pi} \frac{x dx}{1+\sin x}$
- Q13: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 13. $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^7 x dx$
- Q14: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 14. $\int_{0}^{2\pi} \cos^5 x dx$
- Q15: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 15. $\int_{0}^{\frac{\pi}{2}} \frac{\sin x - \cos x}{1+\sin x \cos x} dx$
- Q16: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 16. $\int_{0}^{\pi} \log(1+\cos x) dx$
- Q17: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 17. $\int_{0}^{a} \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a-x}} dx$
- Q18: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 18. $\int_{0}^{4} |x-1| dx$
- Q19: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 19. Show that $\int_{0}^{a} f(x) g(x) dx = 2 \int_{0}^{a} f(x) dx$, if $f$ and $g$ are defined as $f(x) = f(a-x)$ and $g(x) + g(a-x) = 4$
- Q20: Choose the correct answer in Exercises 20 and 21. 20. The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (x^3 + x \cos x + \tan^5 x + 1) dx$ is
- Q21: Choose the correct answer in Exercises 20 and 21. 21. The value of $\int_{0}^{\frac{\pi}{2}} \log (\frac{4+3\sin x}{4+3\cos x}) dx$ is
CBSE Solutions for Class 12 Mathematics Integrals
- Integrals Class 12 EXERCISE 7.1
- 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 7.8
- Integrals Class 12 EXERCISE 7.9
- Integrals Class 12 Miscellaneous Exercise on Chapter 7
Chapters in CBSE - Class 12 Mathematics
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