일차함수의 정적분 관계

$\displaystyle \int _{- 1} ^{1} {\left\{ f ( x ) \right\} ^{2}} dx = \int _{- 1} ^{1} {( x + 1 ) ^{2}} dx$ $\displaystyle = \int _{- 1} ^{1} {( x ^{2} + 2 x + 1 )} dx$$\displaystyle = 2 \int _{0} ^{1} {( x ^{2} + 1 )} dx$ $\displaystyle = 2 \left[ \frac{1}{3} x ^{3} + x \right] _{0} ^{1}$$\displaystyle = \frac{8}{3}$ $\displaystyle k \left( \int _{- 1} ^{1} {f ( x )} dx \right) ^{2} = k \left( \int _{- 1} ^{1} {( x + 1 )} dx \right) ^{2}$ $\displaystyle = k \left( 2 \int _{0} ^{1} {1} dx \right) ^{2} = 4 k \left[ x \right] _{0} ^{1} = 4 k$ 주어진 등식에서 $\displaystyle \frac{8}{3} = 4 k$ $\displaystyle \therefore k = \frac{2}{3}$