로그 관계식과 부분적분

$\displaystyle f \left( x \right) = \ln \left\{ \frac{g \left( x \right)}{1 + xf' \left( x \right)} \right\}$에서 $g \left( x \right) = e ^{f \left( x \right)} \left\{ 1 + xf' \left( x \right) \right\} = e ^{f \left( x \right)} + xf' \left( x \right) e ^{f \left( x \right)}$ $\displaystyle \int _{1} ^{2} {g \left( x \right) dx} = 34$에서 $\displaystyle \int _{1} ^{2} {\left\{ e ^{f \left( x \right)} + xf' \left( x \right) e ^{f \left( x \right)} \right\} dx} = 34$ $\displaystyle \int _{1} ^{2} {e ^{f \left( x \right)} dx} + \int _{1} ^{2} {xf' \left( x \right) e ^{f \left( x \right)} dx} = 34$ $\displaystyle \int _{1} ^{2} {e ^{f \left( x \right)} dx} + \left[ \begin{array}{l} xe ^{f \left( x \right)} \begin{matrix} \\ \end{matrix} \end{array} \right] _{1} ^{2} - \int _{1} ^{2} {e ^{f \left( x \right)} dx} = 34$ $2 e ^{f \left( 2 \right)} - e ^{f \left( 1 \right)} = 34$ $f \left( 1 \right) = 4 \ln 2$이므로 $e ^{f \left( 2 \right)} = 25$ $\displaystyle \int _{1} ^{2} {xg \left( x \right) dx} = 53$에서 $\displaystyle \int _{1} ^{2} {\left\{ xe ^{f \left( x \right)} + x ^{2} f' \left( x \right) e ^{f \left( x \right)} \right\} dx} = 53$ $\displaystyle \int _{1} ^{2} {xe ^{f \left( x \right)} dx} + \int _{1} ^{2} {x ^{2} f' \left( x \right) e ^{f \left( x \right)} dx} = 53$ $\displaystyle \int _{1} ^{2} {xe ^{f \left( x \right)} dx} + \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} x ^{2} e ^{f \left( x \right)} \end{array} \right] _{1} ^{2} - \int _{1} ^{2} {2 xe ^{f \left( x \right)} dx} = 53$ $\displaystyle \therefore \int _{1} ^{2} {xe ^{f \left( x \right)} dx}$$= \left\{ 4 e ^{f \left( 2 \right)} - e ^{f \left( 1 \right)} \right\} - 53$ $= 100 - 16 - 53$$= 31$