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Q8:
By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19.
8. $\int_{0}^{\frac{\pi}{4}} \log(1+\tan x) dx$
Solution :
Step 1: Identifying the Definite Integral and Applying Property
Let the given integral be denoted as $I$:
$I = \int_{0}^{\frac{\pi}{4}} \log(1+\tan x) \, dx$ --- (Equation 1)
We utilize the King's Property of definite integrals, which states that for a function $f(x)$ continuous on $[a, b]$:
$\int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a + b - x) \, dx$ [By Property of Definite Integrals]
Here, $a = 0$ and $b = \frac{\pi}{4}$. Thus, $a + b - x = \frac{\pi}{4} - x$.
Step 2: Substitution and Simplification
Substituting $x \to \frac{\pi}{4} - x$ into Equation 1:
$I = \int_{0}^{\frac{\pi}{4}} \log\left(1 + \tan\left(\frac{\pi}{4} - x\right)\right) \, dx$
Applying the trigonometric identity $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$:
$\tan\left(\frac{\pi}{4} - x\right) = \frac{\tan(\frac{\pi}{4}) - \tan x}{1 + \tan(\frac{\pi}{4})\tan x} = \frac{1 - \tan x}{1 + \tan x}$ [Since $\tan(\frac{\pi}{4}) = 1$]
Substitute this into the integral:
$I = \int_{0}^{\frac{\pi}{4}} \log\left(1 + \frac{1 - \tan x}{1 + \tan x}\right) \, dx$
Step 3: Algebraic Reduction
Combine the terms inside the logarithm:
$1 + \frac{1 - \tan x}{1 + \tan x} = \frac{1 + \tan x + 1 - \tan x}{1 + \tan x} = \frac{2}{1 + \tan x}$
Therefore, the integral becomes:
$I = \int_{0}^{\frac{\pi}{4}} \log\left(\frac{2}{1 + \tan x}\right) \, dx$
Using the logarithmic property $\log(\frac{A}{B}) = \log A - \log B$:
$I = \int_{0}^{\frac{\pi}{4}} (\log 2 - \log(1 + \tan x)) \, dx$
Step 4: Solving for I
Split the integral:
$I = \int_{0}^{\frac{\pi}{4}} \log 2 \, dx - \int_{0}^{\frac{\pi}{4}} \log(1 + \tan x) \, dx$
Observing that the second term is the original integral $I$ (from Equation 1):
$I = \int_{0}^{\frac{\pi}{4}} \log 2 \, dx - I$
$2I = \log 2 \cdot [x]_{0}^{\frac{\pi}{4}}$
$2I = \log 2 \cdot \left(\frac{\pi}{4} - 0\right)$
$2I = \frac{\pi}{4} \log 2$
$I = \frac{\pi}{8} \log 2$
Final Solution: $\frac{\pi}{8} \log 2$
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. $\int_{0}^{1} x(1-x)^n dx$
- 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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