integral from 0 to ln(3) of 2(3-xe^x)

解

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

4

解答ステップ

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

定数を除く:

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

= 2 \cdot \int_{0}^{l n (3)} 3 - 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

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

\int_{0}^{l n (3)} 3 d x = 3 ln (3)

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

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

簡素化

= 4