csat-suneung 2022 Q30 (Calculus)

csat-suneung · South-Korea · csat__math 4 marks Integration by Parts Definite Integral Evaluation by Parts

A function $f ( x )$ that is increasing and differentiable on the set of all real numbers satisfies the following conditions. (가) $f ( 1 ) = 1 , \int _ { 1 } ^ { 2 } f ( x ) d x = \frac { 5 } { 4 }$ (나) When the inverse function of $f ( x )$ is $g ( x )$, for all real numbers $x \geq 1$, $g ( 2 x ) = 2 f ( x )$. When $\int _ { 1 } ^ { 8 } x f ^ { \prime } ( x ) d x = \frac { q } { p }$, find the value of $p + q$. (Given that $p$ and $q$ are coprime natural numbers.) [4 points]

A function $f ( x )$ that is increasing and differentiable on the set of all real numbers satisfies the following conditions.\\
(가) $f ( 1 ) = 1 , \int _ { 1 } ^ { 2 } f ( x ) d x = \frac { 5 } { 4 }$\\
(나) When the inverse function of $f ( x )$ is $g ( x )$, for all real numbers $x \geq 1$, $g ( 2 x ) = 2 f ( x )$.\\
When $\int _ { 1 } ^ { 8 } x f ^ { \prime } ( x ) d x = \frac { q } { p }$, find the value of $p + q$. (Given that $p$ and $q$ are coprime natural numbers.) [4 points]

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