로그곡선 직사각형 넓이 적분

$\mathrm{Q} \mathit{\left( t , 0 \right)}$, $\mathrm{R} \mathit{\left( 0 , 2 \ln \left( t + 1 \right) \right)}$이므로 직사각형 $\mathrm{OQPR}$의 넓이는 $f \left( t \right) = 2 t \ln \left( t + 1 \right)$ 따라서 $\displaystyle \int _{1} ^{3} {f \left( t \right)} dt$ $\displaystyle = \int _{1} ^{3} {\left\{ 2 t \ln \left( t + 1 \right) \right\} dt}$ $\displaystyle = \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} t ^{2} \ln \left( t + 1 \right) \end{array} \right] _{1} ^{3} - \int _{1} ^{3} {\frac{t ^{2}}{t + 1}} dt$ $\displaystyle = \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} t ^{2} \ln \left( t + 1 \right) \end{array} \right] _{1} ^{3} - \int _{1} ^{3} {\left( t - 1 + \frac{1}{t + 1} \right) dt}$ $\displaystyle = \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} t ^{2} \ln \left( t + 1 \right) \end{array} \right] _{1} ^{3} - \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} \frac{1}{2} t ^{2} - t + \ln \left( t + 1 \right) \end{array} \right] _{1} ^{3}$ $\displaystyle = \left( 9 \ln 4 - \ln 2 \right) - \left( \frac{9}{2} - 3 + \ln 4 - \frac{1}{2} + 1 - \ln 2 \right)$ $= - 2 + 16 \ln 2$