Question

The integral $\int\limits^{4}_{2}$ $\frac{log\,x^{2}}{log\,x^{2}+log\left(36-12x+x^{2}\right)}dx$ is equal to

• A. 2
• B. 4
• C. 1
• D. 6

Answer 1

Answer: (C)

$I =\int\limits_{2}^{4} \frac{\log x ^{2}}{\log x ^{2}+\log ( x -6)^{2}} dx \ldots .(1)$
using $\int\limits_{a}^{b} f ( x ) d x =\int\limits_{ a }^{ b } f ( a + b - x ) d x$
$I =\int\limits_{2}^{4} \frac{\log (6- x )^{2}}{\log (6- x )^{2}+\log x ^{2}} dx \ldots(2)$
$(1)+(2)$ gives
$2 I =\int\limits_{2}^{4} 1 \,dx =2$
$I =1$