연립 적분방정식

$\displaystyle \int _{0} ^{1} \left\{ f ( t ) + g ( t ) \right\} dt = C _{1}$, $\displaystyle \int _{0} ^{1} \left\{ f ( t ) - g ( t ) \right\} dt = C _{2}$라 하면 주어진 조건에 의해서 $f ( x ) = 2 x + C _{1}$, $g ( x ) = 3 x ^{2} + C _{2}$ $f ( x ) + g ( x ) = 3 x ^{2} + 2 x + C _{1} + C _{2}$이므로 $\displaystyle \int _{0} ^{1} \left\{ f ( t ) + g ( t ) \right\} dt$ $\displaystyle = \int _{0} ^{1} \left\{ 3 t ^{2} + 2 t + C _{1} + C _{2} \right\} dt$ $= \left[ t ^{3} + t ^{2} + ( C _{1} + C _{2} ) t \right] _{0} ^{1}$ $= C _{1} + C _{2} + 2$$= C _{1}$ ∴ $C _{2} = - 2$ $\displaystyle \int _{0} ^{1} \left\{ f ( t ) - g ( t ) \right\} dt$ $\displaystyle = \int _{0} ^{1} \left\{ 3 t ^{2} - 2 t + C _{1} - C _{2} \right\} dt$ $= \left[ t ^{3} - t ^{2} + ( C _{1} - C _{2} ) t \right] _{0} ^{1}$ $= C _{1} - C _{2}$$= C _{2}$ ∴ $C _{1} = 2 C _{2} = - 4$ ∴ $f ( x ) = 2 x - 4$, $g ( x ) = 3 x ^{2} - 2$ ∴ $f ( 1 ) + g ( 2 )$$= ( - 2 ) + ( 10 ) = 8$