int_{2}^{4} frac{log(x^2)}{log(x^2) + log(36 | vedclass

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(D) Let $I = \int_{2}^{4} \frac{\log(x^2)}{\log(x^2) + \log((6-x)^2)} \, dx$. Using the property $\log(a^2) = 2\log|a|$,we have: $I = \int_{2}^{4} \fr

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$1$

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(D) Let $I = \int_{2}^{4} \frac{\log(x^2)}{\log(x^2) + \log((6-x)^2)} \, dx$. Using the property $\log(a^2) = 2\log|a|$,we have: $I = \int_{2}^{4} \frac{2\log x}{2\log x + 2\log(6-x)} \, dx = \int_{2}^{4} \frac{\log x}{\log x + \log(6-x)} \, dx \quad \dots(1)$. Using the property $\int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a+b-x) \, dx$,where $a+b = 2+4 = 6$: $I = \int_{2}^{4} \frac{\log(6-x)}{\log(6-x) + \log(6-(6-x))} \, dx = \int_{2}^{4} \frac{\log(6-x)}{\log(6-x) + \log x} \, dx \quad \dots(2)$. Adding $(1)$ and $(2)$: $2I = \int_{2}^{4} \frac{\log x + \log(6-x)}{\log x + \log(6-x)} \, dx = \int_{2}^{4} 1 \, dx$. $2I = [x]_{2}^{4} = 4 - 2 = 2$. $I = 1$.

$\int_{2}^{4} \frac{\log(x^2)}{\log(x^2) + \log(36 - 12x + x^2)} \, dx = $

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