integral from 0 to ln(4) of 3(4-xe^x)

解

\int_{0}^{l n (4)} 3 (4 - x e^{x}) d x

9

解答ステップ

\int_{0}^{l n (4)} 3 (4 - x e^{x}) d x

定数を除く:

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

= 3 \cdot \int_{0}^{l n (4)} 4 - x 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

= 3 (\int_{0}^{l n (4)} 4 d x - \int_{0}^{l n (4)} x e^{x} d x)

\int_{0}^{l n (4)} 4 d x = 8 ln (2)

\int_{0}^{l n (4)} x e^{x} d x = 8 ln (2) - 3

= 3 (8 ln (2) - (8 ln (2) - 3))

簡素化

= 9