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Q10:
By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19.
10. $\int_{0}^{\frac{\pi}{2}} (2\log \sin x - \log \sin 2x) dx$
Solution :
Step 1: Simplifying the Integrand
The given integral is $I = \int_{0}^{\frac{\pi}{2}} (2\log \sin x - \log \sin 2x) \, dx$.
Using the property of logarithms $n\log a = \log a^n$, we can rewrite the first term. Using the double angle formula for sine, $\sin 2x = 2 \sin x \cos x$, we expand the second term:
$I = \int_{0}^{\frac{\pi}{2}} (\log \sin^2 x - \log(2 \sin x \cos x)) \, dx$
Applying the quotient rule for logarithms, $\log a - \log b = \log(\frac{a}{b})$:
$I = \int_{0}^{\frac{\pi}{2}} \log\left(\frac{\sin^2 x}{2 \sin x \cos x}\right) \, dx$
$I = \int_{0}^{\frac{\pi}{2}} \log\left(\frac{\sin x}{2 \cos x}\right) \, dx$
$I = \int_{0}^{\frac{\pi}{2}} \log\left(\frac{1}{2} \tan x\right) \, dx$
Step 2: Decomposition using Logarithmic Properties
Using the property $\log(ab) = \log a + \log b$:
$I = \int_{0}^{\frac{\pi}{2}} (\log(2^{-1}) + \log \tan x) \, dx$
$I = \int_{0}^{\frac{\pi}{2}} -\log 2 \, dx + \int_{0}^{\frac{\pi}{2}} \log \tan x \, dx$
Step 3: Evaluating the Separate Integrals
Part A: $\int_{0}^{\frac{\pi}{2}} -\log 2 \, dx$
$= -\log 2 [x]_{0}^{\frac{\pi}{2}} = -\log 2 (\frac{\pi}{2} - 0) = -\frac{\pi}{2} \log 2$
Part B: $\int_{0}^{\frac{\pi}{2}} \log \tan x \, dx$
Let $J = \int_{0}^{\frac{\pi}{2}} \log \tan x \, dx$. By the property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$ [King's Property]:
$J = \int_{0}^{\frac{\pi}{2}} \log \tan(\frac{\pi}{2} - x) \, dx = \int_{0}^{\frac{\pi}{2}} \log \cot x \, dx$
$J = \int_{0}^{\frac{\pi}{2}} \log(\frac{1}{\tan x}) \, dx = \int_{0}^{\frac{\pi}{2}} -\log \tan x \, dx = -J$
Therefore, $2J = 0 \implies J = 0$.
Step 4: Final Summation
$I = (-\frac{\pi}{2} \log 2) + 0$
$I = -\frac{\pi}{2} \log 2$
Final Solution: -\frac{\pi}{2} \log 2
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. $\int_{0}^{\frac{\pi}{2}} \cos^2 x dx$
- Q2: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 2. $\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx$
- 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. $\int_{0}^{2} x \sqrt{2-x} dx$
- 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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