integral of x^8xe^{x^2}

解答

\int x^{8} x e^{x^{2}} d x

\frac{1}{2} e^{x^{2}} x^{8} - 2 e^{x^{2}} x^{6} + 6 e^{x^{2}} x^{4} - 12 e^{x^{2}} x^{2} + 12 e^{x^{2}} + C

求解步骤

\int x^{8} x e^{x^{2}} d x

化简

x^{8} x : x^{9}

= \int x^{9} e^{x^{2}} d x

使用分布积分法

= \frac{1}{10} e^{x^{2}} x^{10} - \int \frac{1}{5} e^{x^{2}} x^{11} d x

\int \frac{1}{5} e^{x^{2}} x^{11} d x = \frac{1}{10} (e^{x^{2}} x^{10} - 5 (e^{x^{2}} x^{8} - 4 (e^{x^{2}} x^{6} - 3 (e^{x^{2}} x^{4} - 2 (e^{x^{2}} x^{2} - e^{x^{2}})))))

= \frac{1}{10} e^{x^{2}} x^{10} - \frac{1}{10} (e^{x^{2}} x^{10} - 5 (e^{x^{2}} x^{8} - 4 (e^{x^{2}} x^{6} - 3 (e^{x^{2}} x^{4} - 2 (e^{x^{2}} x^{2} - e^{x^{2}})))))

化简

\frac{1}{10} e^{x^{2}} x^{10} - \frac{1}{10} (e^{x^{2}} x^{10} - 5 (e^{x^{2}} x^{8} - 4 (e^{x^{2}} x^{6} - 3 (e^{x^{2}} x^{4} - 2 (e^{x^{2}} x^{2} - e^{x^{2}}))))) : \frac{1}{2} e^{x^{2}} x^{8} - 2 e^{x^{2}} x^{6} + 6 e^{x^{2}} x^{4} - 12 e^{x^{2}} x^{2} + 12 e^{x^{2}}

= \frac{1}{2} e^{x^{2}} x^{8} - 2 e^{x^{2}} x^{6} + 6 e^{x^{2}} x^{4} - 12 e^{x^{2}} x^{2} + 12 e^{x^{2}}

解答补常数

= \frac{1}{2} e^{x^{2}} x^{8} - 2 e^{x^{2}} x^{6} + 6 e^{x^{2}} x^{4} - 12 e^{x^{2}} x^{2} + 12 e^{x^{2}} + C

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