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人気のある微分積分 >

integral of 2(x+20)e^{x/(20)}

解

\int 2 (x + 20) e^{\frac{x}{20}} d x

解

40 e^{\frac{x}{20}} x + C

解答ステップ

\int 2 (x + 20) e^{\frac{x}{20}} d x

定数を除く:

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

= 2 \cdot \int (x + 20) e^{\frac{x}{20}} d x

拡張

(x + 20) e^{\frac{x}{20}} : e^{\frac{x}{20}} x + 20 e^{\frac{x}{20}}

= 2 \cdot \int e^{\frac{x}{20}} x + 20 e^{\frac{x}{20}} 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 e^{\frac{x}{20}} x d x + \int 20 e^{\frac{x}{20}} d x)

\int e^{\frac{x}{20}} x d x = 20 e^{\frac{x}{20}} x - 400 e^{\frac{x}{20}}

\int 20 e^{\frac{x}{20}} d x = 400 e^{\frac{x}{20}}

= 2 (20 e^{\frac{x}{20}} x - 400 e^{\frac{x}{20}} + 400 e^{\frac{x}{20}})

簡素化

= 40 e^{\frac{x}{20}} x

定数を解答に追加する

= 40 e^{\frac{x}{20}} x + C

グラフ

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