Question

Area bounded by the curve $y=\log (x-2)$ $x$-axis and $x=4$ is equal to

• A. $2\log 2+1$
• B. $\log 2-1$
• C. $\log 2+1$
• D. $1-2\log 2$
• E. $2\log 2-1$

Answer 1

Answer: (E)

$2\log 2-1$

Given curve is
$y=\log (x-2)$.
Thus required area $={\int }_{x=3}^{4}ydx$
$={\int }_3^4\log (x-2)\cdot 1dx$
$=[x\cdot \log (x-2{)}_3^4-{\int }_3^4\frac{x}{x-2}dx$
$=4\log 2-{\int }_{3}4\{1+\frac{2}{x-2}\}dx$
$=4\log 2-[x+2\log (x-2){]}_3^4$
$=4\log 2-[4+2\log 2-3]$
$=4\log 2-2\log 2$

$=2\log 2-1$