적분 포함 함수식

$\displaystyle \int _{0} ^{1} tf ( t ) dt = c$라 하면 $f ( x ) = e ^{x} + c$ $c$$\displaystyle = \int _{0} ^{1} {} tf ( t ) dt$$\displaystyle = \int _{0} ^{1} ( te ^{t} + ct ) dt$ $\displaystyle = \left[ \begin{array}{l} t e ^{t} \begin{matrix} \\ \end{matrix} \end{array} \right] _{0} ^{1} - \int _{0} ^{1} e ^{t} dt + \left[ \frac{ct ^{2}}{2} \right] _{0} ^{1}$ $\displaystyle = e - \left[ \begin{array}{l} e ^{t} \begin{matrix} \\ \end{matrix} \end{array} \right] _{0} ^{1} + \frac{c}{2}$ $\displaystyle = 1 + \frac{c}{2}$ ∴ $c = 2$ ∴ $f ( x )$$= e ^{x} + 2$ ∴ $\displaystyle \int _{0} ^{1} f ( x ) dx$$\displaystyle = \int _{0} ^{1} ( e ^{x} + 2 ) dx$ $\displaystyle = \left[ \begin{array}{l} e ^{x} + 2 x \begin{matrix} \\ \end{matrix} \end{array} \right] _{0} ^{1}$ $= e + 1$