integral from 0 to-5 of (2/5 x^2+2x-2)

解

\int_{0}^{- 5} (\frac{2}{5} x^{2} + 2 x - 2) d x

\frac{55}{3}

十進法表記

18.33333 \dots

解答ステップ

\int_{0}^{- 5} \frac{2}{5} x^{2} + 2 x - 2 d x

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{- 5}^{0} \frac{2}{5} x^{2} + 2 x - 2 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

= - (\int_{- 5}^{0} \frac{2}{5} x^{2} d x + \int_{- 5}^{0} 2 x d x - \int_{- 5}^{0} 2 d x)

\int_{- 5}^{0} \frac{2}{5} x^{2} d x = \frac{50}{3}

\int_{- 5}^{0} 2 x d x = - 25

\int_{- 5}^{0} 2 d x = 10

= - (\frac{50}{3} - 25 - 10)

簡素化

- (\frac{50}{3} - 25 - 10) : \frac{55}{3}

= \frac{55}{3}