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人気のある微分積分 >

integral of x^{1/3}e^{x^{1/3}}

解

\int x^{\frac{1}{3}} e^{x^{\frac{1}{3}}} d x

解

3 (e^{x^{\frac{1}{3}}} x - 3 (e^{x^{\frac{1}{3}}} x^{\frac{2}{3}} - 2 (e^{x^{\frac{1}{3}}} x^{\frac{1}{3}} - e^{x^{\frac{1}{3}}}))) + C

解答ステップ

\int x^{\frac{1}{3}} e^{x^{\frac{1}{3}}} d x

u置換積分法を適用する

= \int 3 e^{u} u^{3} d u

定数を除く:

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

= 3 \cdot \int e^{u} u^{3} d u

部分積分を適用する

= 3 (u^{3} e^{u} - \int 3 u^{2} e^{u} d u)

\int 3 u^{2} e^{u} d u = 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

= 3 (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

代用を戻す

u = x^{\frac{1}{3}}

= 3 ((x^{\frac{1}{3}})^{3} e^{x^{\frac{1}{3}}} - 3 ((x^{\frac{1}{3}})^{2} e^{x^{\frac{1}{3}}} - 2 (e^{x^{\frac{1}{3}}} x^{\frac{1}{3}} - e^{x^{\frac{1}{3}}})))

簡素化

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

= 3 (e^{x^{\frac{1}{3}}} x - 3 (e^{x^{\frac{1}{3}}} x^{\frac{2}{3}} - 2 (e^{x^{\frac{1}{3}}} x^{\frac{1}{3}} - e^{x^{\frac{1}{3}}})))

定数を解答に追加する

= 3 (e^{x^{\frac{1}{3}}} x - 3 (e^{x^{\frac{1}{3}}} x^{\frac{2}{3}} - 2 (e^{x^{\frac{1}{3}}} x^{\frac{1}{3}} - e^{x^{\frac{1}{3}}}))) + C

グラフ

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