In a mathematics class where the topic of integration by parts is being taught, Teacher Ebru writes on the board $$\int u d v = u v - \int v d u$$ rule. Then, in solving a problem, Mehmet chooses functions $f ( x )$ and $g ( x )$ in place of u and v respectively and applies this rule to obtain $$\int f ( x ) g ^ { \prime } ( x ) d x = \frac { f ( x ) } { x } - \int \frac { 2 } { x ^ { 2 } } d x$$ equality. Given that $\mathbf { f } ( \mathbf { 1 } ) = \mathbf { 2 }$, what is the value of $\mathbf { f } ( \mathbf { e } )$? A) 2 B) 4 C) 6 D) 8 E) 10
In a mathematics class where the topic of integration by parts is being taught, Teacher Ebru writes on the board
$$\int u d v = u v - \int v d u$$
rule. Then, in solving a problem, Mehmet chooses functions $f ( x )$ and $g ( x )$ in place of u and v respectively and applies this rule to obtain
$$\int f ( x ) g ^ { \prime } ( x ) d x = \frac { f ( x ) } { x } - \int \frac { 2 } { x ^ { 2 } } d x$$
equality.
Given that $\mathbf { f } ( \mathbf { 1 } ) = \mathbf { 2 }$, what is the value of $\mathbf { f } ( \mathbf { e } )$?
A) 2
B) 4
C) 6
D) 8
E) 10