Question

Evaluate the integral: \[ \int_{5}^{9} \frac{\log 3x^2}{\log 3x^2 + \log (588 - 84x + 3x^2)} dx \]

• A. 2
• B. 1
• C. \( \frac{1}{2} \)
• D. 4

Hint:

Use logarithmic properties and symmetry of definite integrals to simplify complex integrals.

Answer 1

The Correct Option is A

{Let } \[I = \int_{5}^{9} \frac{\log 3x^2}{\log 3x^2 + \log (588 - 84x + 3x^2)} \, dx \quad \cdots (i)\] We can rewrite the second term in the denominator as follows:
\[\log (588 - 84x + 3x^2) = \log (3(196 - 28x + x^2)) = \log (3(14-x)^2)\] \[{Now, using the property of definite integrals, } \int_a^b f(x) \, dx = \int_a^b f(a+b-x) \, dx, { we have:}\] \[I = \int_{5}^{9} \frac{\log (3(14-x)^2)}{\log (3(14-x)^2) + \log (3x^2)} \, dx\] \[I = \int_{5}^{9} \frac{\log 3 + 2\log (14-x)}{\log 3 + 2\log (14-x) + \log 3 + 2\log x} \, dx\] \[I = \int_{5}^{9} \frac{\log 3 + 2\log (14-x)}{2\log 3 + 2\log (14-x) + 2\log x} \, dx\] \[I = \int_{5}^{9} \frac{\log 3 + 2\log (14-x)}{2(\log 3 + \log (14-x) + \log x)} \, dx\] \[I = \int_{5}^{9} \frac{\log 3 + 2\log (14-x)}{2\log (3x(14-x))} \, dx \quad \cdots (ii)\] \[{Adding equations (i) and (ii):}\] \[2I = \int_{5}^{9} \frac{\log 3x^2 + \log (3(14-x)^2)}{\log 3x^2 + \log (3(14-x)^2)} \, dx\] \[2I = \int_{5}^{9} 1 \, dx\] \[2I = [x]_5^9\] \[2I = 9 - 5 = 4\] I = 2

Answer 2

Step 1: Simplify the given integral
We are asked to evaluate the integral: \[ I = \int_{5}^{9} \frac{\log 3x^2}{\log 3x^2 + \log (588 - 84x + 3x^2)} dx \] Notice that the denominator contains two logarithmic terms, which can be combined using the logarithmic identity: \[ \log a + \log b = \log(ab) \] So the integral becomes: \[ I = \int_{5}^{9} \frac{\log 3x^2}{\log \left( 3x^2 (588 - 84x + 3x^2) \right)} dx \]
Step 2: Simplify the expression inside the logarithm
Let's examine the expression inside the logarithm: \[ 3x^2 (588 - 84x + 3x^2) \] Expanding the terms: \[ 3x^2(588 - 84x + 3x^2) = 1764x^2 - 252x^3 + 9x^4 \] So the integral now becomes: \[ I = \int_{5}^{9} \frac{\log 3x^2}{\log (9x^4 - 252x^3 + 1764x^2)} dx \]
Step 3: Symmetry observation
Notice that the numerator and denominator have a certain symmetry. A natural guess is that the numerator and denominator might cancel out in such a way that simplifies the integral. Through detailed calculation or numerical methods, you will find that the value of this integral simplifies to 2. Thus, the final answer is: \[ \boxed{2} \]