부분적분으로 함숫값

$\displaystyle \int _{0} ^{1} {\begin{aligned} f ( x ) g' ( x ) dx \\ \end{aligned}}$ $\displaystyle = \left[ f ( x ) g ( x ) \right] _{0} ^{1} - \int _{0} ^{1} {f' ( x )} g ( x ) dx$ $\displaystyle = f ( 1 ) g ( 1 ) - f ( 0 ) g ( 0 ) - \int _{0} ^{1} {\frac{x ^{2}}{( 1 + x ^{3} ) ^{2}}} dx$ $g ( 1 ) = 1 , g ( 0 ) = 0$이고 $1 + x ^{3} = t$로 치환하면 $\displaystyle f ( 1 ) - \frac{1}{3} \int _{1} ^{2} {\frac{1}{t ^{2}} dt}$$\displaystyle = f ( 1 ) - \frac{1}{6}$ $\displaystyle f ( 1 ) - \frac{1}{6} = \frac{1}{6}$이므로 $\displaystyle f ( 1 ) = \frac{1}{3}$