Question

A differentiable function $f(x)$ satisfying integral equation $f(x)={\left({\log }_{c}x\right)}^2-\underset{1}{\overset{e}{\int }}\frac{f(t)}{t}dt$. Suppose $f(x)+\frac{1}{6}=g(x)$.
The area of the region bounded by the tangent line of $y=f(x)$ at the point $(e,f(e))$, the curve $y$ $=f(x)$ and the line $x=1$ is equal to

• A. $e-\frac{1}{e}+2$
• B. $e+\frac{1}{e}-3$
• C. $e-\frac{1}{e}+3$
• D. $e+\frac{1}{e}-2$

Answer 1

Answer: (B)

$f(x)=(\ln x{)}^2-\frac{1}{6}$
$f^{\prime }(x)=\frac{2\ln x}{x}>0\text{ for }x\in (1,e)$
$y=\frac{2x}{e}-\frac{7}{6}$
$A=\underset{1}{\overset{e}{\int }}\left((\ln x{)}^2-\frac{1}{6}\right)-\left(\frac{2x}{e}-\frac{7}{6}\right)dx$
$A=e+\frac{1}{e}-3$