정적분을 포함한 함수식

$\displaystyle f ( x ) = e ^{x} {}^{2} + \int _{0} ^{1} t f ( t ) dt$에서 $\displaystyle \int _{0} ^{1} t f ( t ) dt = a$라 하면 $f ( x ) = e ^{x ^{2}} + a$이므로 $\displaystyle a = \int _{0} ^{1} {t \cdot f ( t )} dt$$\displaystyle = \int _{0} ^{1} {t \left( e ^{t ^{2}} + a \right)} dt$ $\displaystyle = \int _{0} ^{1} {\left( t \cdot e ^{t ^{2}} + at \right)} dt$$\displaystyle = \left[ \frac{1}{2} e ^{t ^{2}} + \frac{1}{2} at ^{2} \right] _{0} ^{1}$ $\displaystyle = \frac{1}{2} e + \frac{1}{2} a - \frac{1}{2}$ $\displaystyle a = \frac{1}{2} e + \frac{1}{2} a - \frac{1}{2}$ $\therefore a = e - 1$