접선 거리로 정의된 함수의 적분

$\displaystyle f \left( x \right) = \frac{1}{2} x ^{2} - x + \ln \left( 1 + x \right)$에서 $\displaystyle f' \left( x \right) = x - 1 + \frac{1}{1 + x} = \frac{\left( x - 1 \right) \left( x + 1 \right) + 1}{x + 1} = \frac{x ^{2}}{x + 1}$ 곡선 $y = f \left( x \right)$ 위의 점 $\left( s , f \left( s \right) \right)$에서의 접선의 방정식은 $y - f \left( s \right) = f' \left( s \right) \left( x - s \right)$ 이고 $x = 0$을 대입하면 $y - f \left( s \right) = - sf' \left( s \right)$, 즉 $y = f \left( s \right) - s f' \left( s \right)$ 이므로 이 접선이 $y$축과 만나는 점의 $y$좌표는 $f \left( s \right) - sf' \left( s \right)$ 한편 점 $\left( s , f \left( s \right) \right)$에서 $y$축에 내린 수선의 발의 $y$좌표는 $f \left( s \right)$이고 두 점 $\left( 0 , f \left( s \right) \right)$, $\left( 0 , f \left( s \right) - sf' \left( s \right) \right)$ 사이의 거리가 $t$이므로 $t$$= \left| f \left( s \right) - \left\{ f \left( s \right) - sf' \left( s \right) \right\} \right|$ $= \left| sf' \left( s \right) \right|$$\displaystyle = \left| s \cdot \frac{s ^{2}}{s + 1} \right|$$\displaystyle = \left| \frac{s ^{3}}{s + 1} \right|$ $\displaystyle = \frac{s ^{3}}{s + 1} \left( \because s > 0 \right)$ 이때 $s = g \left( t \right)$이고 $\displaystyle \frac{dt}{ds} = \frac{3 s ^{2} \left( s + 1 \right) - s ^{3}}{\left( s + 1 \right) ^{2}} = \frac{2 s ^{3} + 3 s ^{2}}{\left( s + 1 \right) ^{2}}$ $\displaystyle t = \frac{1}{2}$일 때, $\displaystyle \frac{s ^{3}}{s + 1} = \frac{1}{2}$에서 $2 s ^{3} - s - 1 = 0$, $\left( s - 1 \right) \left( 2 s ^{2} + 2 s + 1 \right) = 0$ $\therefore$ $s = 1 \left( \because 2 s ^{2} + 2 s + 1 > 0 \right)$ $\displaystyle t = \frac{27}{4}$일 때, $\displaystyle \frac{s ^{3}}{s + 1} = \frac{27}{4}$에서 $4 s ^{3} - 27 s - 27 = 0$, $\left( s - 3 \right) \left( 4 s ^{2} + 12 s + 9 \right) = 0$ $\left( s - 3 \right) \left( 2 s + 3 \right) ^{2} = 0$ $\therefore$ $s = 3 \left( \because s > 0 \right)$ $\therefore$ $\displaystyle \int _{\frac{1}{2}} ^{\frac{27}{4}} {} g \left( t \right) dt$ $\displaystyle = \int _{1} ^{3} {} s \cdot \frac{2 s ^{3} + 3 s ^{2}}{\left( s + 1 \right) ^{2}} ds$ $\displaystyle = \int _{1} ^{3} {} \frac{2 s ^{4} + 3 s ^{3}}{\left( s + 1 \right) ^{2}} ds$ $\displaystyle = \int _{1} ^{3} {} \frac{2 s ^{2} \left( s ^{2} + 2 s + 1 \right) - s ^{3} - 2 s ^{2}}{\left( s + 1 \right) ^{2}} ds$ $\displaystyle = \int _{1} ^{3} {} \left\{ 2 s ^{2} - \frac{s \left( s ^{2} + 2 s + 1 \right) - s}{\left( s + 1 \right) ^{2}} \right\} ds$ $\displaystyle = \int _{1} ^{3} {} \left\{ 2 s ^{2} - s + \frac{s}{\left( s + 1 \right) ^{2}} \right\} ds$ $\displaystyle = \int _{1} ^{3} {} \left\{ 2 s ^{2} - s + \frac{1}{s + 1} - \frac{1}{\left( s + 1 \right) ^{2}} \right\} ds$ $\displaystyle = \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} \frac{2}{3} s ^{3} - \frac{1}{2} s ^{2} + \ln \left| s + 1 \right| + \frac{1}{s + 1} \end{array} \right] _{1} ^{3}$ $\displaystyle = \left( 18 - \frac{9}{2} + \ln 4 + \frac{1}{4} \right) - \left( \frac{2}{3} - \frac{1}{2} + \ln 2 + \frac{1}{2} \right)$ $\displaystyle = \left( \frac{55}{4} + 2 \ln 2 \right) - \left( \frac{2}{3} + \ln 2 \right)$ $\displaystyle = \frac{157}{12} + \ln 2$