integral of xe^{2/x}

解答

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

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

求解步骤

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

使用分布积分法

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

\int - e^{\frac{2}{x}} d x = - e^{\frac{2}{x}} x + 2 Ei (\frac{2}{x})

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

化简

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

解答补常数

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

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