integral of 1/2 xe^{x/6}

解

\int \frac{1}{2} x e^{\frac{x}{6}} d x

3 e^{\frac{x}{6}} (x - 6) + C

解答ステップ

\int \frac{1}{2} x e^{\frac{x}{6}} d x

定数を除く:

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

= \frac{1}{2} \cdot \int x e^{\frac{x}{6}} d x

部分積分を適用する

: 6 e^{\frac{x}{6}} x - \int 6 e^{\frac{x}{6}} d x

= \frac{1}{2} (6 e^{\frac{x}{6}} x - \int 6 e^{\frac{x}{6}} d x)

\int 6 e^{\frac{x}{6}} d x = 36 e^{\frac{x}{6}}

= \frac{1}{2} (6 e^{\frac{x}{6}} x - 36 e^{\frac{x}{6}})

簡素化

\frac{1}{2} (6 e^{\frac{x}{6}} x - 36 e^{\frac{x}{6}}) : 3 e^{\frac{x}{6}} (x - 6)

= 3 e^{\frac{x}{6}} (x - 6)

定数を解答に追加する

= 3 e^{\frac{x}{6}} (x - 6) + C