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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(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 = $

Similar Questions

Let $I(R) = \int_0^R e^{-R \sin x} dx$, where $R > 0$. Then,

$\int\limits_a^b [x] \,dx + \int\limits_a^b [-x] \,dx$,where $[.]$ denotes the greatest integer function,is equal to:

If $\int_0^\pi \frac{d x}{1+2 \sin ^2 x}=k$,then the greatest integer less than or equal to $k$ is

Evaluate the definite integral: $\int_{0}^{\infty} \frac{x}{(1 + x)(1 + x^2)} \, dx$

Evaluate the definite integral: $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{d x}{1+\sqrt{\tan x}}$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module