integral of 5xe^{x/5}

解

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

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

解答ステップ

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

定数を除く:

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

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

部分積分を適用する

= 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}}

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

定数を解答に追加する

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