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人気のある微分積分 >

integral of x^2e^{sqrt(x)}

解

\int x^{2} e^{x} d x

解

2 (e^{x} x^{\frac{5}{2}} - 5 (e^{x} x^{2} - 4 (e^{x} x^{\frac{3}{2}} - 3 (e^{x} x - 2 (e^{x} x - e^{x}))))) + C

解答ステップ

\int x^{2} e^{x} d x

u置換積分法を適用する

= \int 2 e^{u} u^{5} d u

定数を除く:

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

= 2 \cdot \int e^{u} u^{5} d u

部分積分を適用する

= 2 (u^{5} e^{u} - \int 5 u^{4} e^{u} d u)

\int 5 u^{4} e^{u} d u = 5 (u^{4} e^{u} - 4 (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))))

= 2 (u^{5} e^{u} - 5 (u^{4} e^{u} - 4 (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))))

代用を戻す

u = x

= 2 ((x)^{5} e^{x} - 5 ((x)^{4} e^{x} - 4 ((x)^{3} e^{x} - 3 ((x)^{2} e^{x} - 2 (e^{x} x - e^{x})))))

簡素化

2 ((x)^{5} e^{x} - 5 ((x)^{4} e^{x} - 4 ((x)^{3} e^{x} - 3 ((x)^{2} e^{x} - 2 (e^{x} x - e^{x}))))) : 2 (e^{x} x^{\frac{5}{2}} - 5 (e^{x} x^{2} - 4 (e^{x} x^{\frac{3}{2}} - 3 (e^{x} x - 2 (e^{x} x - e^{x})))))

= 2 (e^{x} x^{\frac{5}{2}} - 5 (e^{x} x^{2} - 4 (e^{x} x^{\frac{3}{2}} - 3 (e^{x} x - 2 (e^{x} x - e^{x})))))

定数を解答に追加する

= 2 (e^{x} x^{\frac{5}{2}} - 5 (e^{x} x^{2} - 4 (e^{x} x^{\frac{3}{2}} - 3 (e^{x} x - 2 (e^{x} x - e^{x}))))) + C

グラフ

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