integral of 1/5 xe^{-x/5}

解答

\int \frac{1}{5} x e^{- \frac{x}{5}} d x

- e^{- \frac{x}{5}} (x + 5) + C

求解步骤

\int \frac{1}{5} x e^{- \frac{x}{5}} d x

提出常数:

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

= \frac{1}{5} \cdot \int x e^{- \frac{x}{5}} d x

使用分布积分法

= \frac{1}{5} (- 5 e^{- \frac{x}{5}} x - \int - 5 e^{- \frac{x}{5}} d x)

\int - 5 e^{- \frac{x}{5}} d x = 25 e^{- \frac{x}{5}}

= \frac{1}{5} (- 5 e^{- \frac{x}{5}} x - 25 e^{- \frac{x}{5}})

化简

\frac{1}{5} (- 5 e^{- \frac{x}{5}} x - 25 e^{- \frac{x}{5}}) : - e^{- \frac{x}{5}} (x + 5)

= - e^{- \frac{x}{5}} (x + 5)

解答补常数

= - e^{- \frac{x}{5}} (x + 5) + C

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