integral from 0 to 7 of 5(1-e^x)

解

\int_{0}^{7} 5 (1 - e^{x}) d x

5 (- e^{7} + 8)

十進法表記

- 5443.16579 \dots

解答ステップ

\int_{0}^{7} 5 (1 - e^{x}) d x

定数を除く:

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

= 5 \cdot \int_{0}^{7} 1 - e^{x} 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

= 5 (\int_{0}^{7} 1 d x - \int_{0}^{7} e^{x} d x)

\int_{0}^{7} 1 d x = 7

\int_{0}^{7} e^{x} d x = e^{7} - 1

= 5 (7 - (e^{7} - 1))

簡素化

= 5 (- e^{7} + 8)