integral from 2 to 4 of 1/27 (x^2-2x+3)

解

\int_{2}^{4} \frac{1}{27} (x^{2} - 2 x + 3) d x

\frac{38}{81}

十進法表記

0.46913 \dots

解答ステップ

\int_{2}^{4} \frac{1}{27} (x^{2} - 2 x + 3) d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{27} \cdot \int_{2}^{4} x^{2} - 2 x + 3 d x

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{27} (\int_{2}^{4} x^{2} d x - \int_{2}^{4} 2 x d x + \int_{2}^{4} 3 d x)

\int_{2}^{4} x^{2} d x = \frac{56}{3}

\int_{2}^{4} 2 x d x = 12

\int_{2}^{4} 3 d x = 6

= \frac{1}{27} (\frac{56}{3} - 12 + 6)

簡素化

\frac{1}{27} (\frac{56}{3} - 12 + 6) : \frac{38}{81}

= \frac{38}{81}