integral from y to 1 of x^2e^{xy}

解答

\int_{y}^{1} x^{2} e^{x y} d x

\frac{e ^{y}}{y} - \frac{2 e ^{y}}{y ^{2}} + \frac{2 e ^{y} + 2 ( y ^{2} e ^{y^{2}} - e ^{y^{2}} )}{y ^{3}} - y e^{y^{2}}

求解步骤

\int_{y}^{1} x^{2} e^{x y} d x

使用分布积分法

= [e^{y x} \frac{1}{y} x^{2} - \int e^{y x} \frac{2}{y} x d x]_{y}^{1}

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

= [e^{y x} \frac{1}{y} x^{2} - \frac{2}{y} (e^{y x} \frac{1}{y} x - e^{y x} \frac{1}{y ^{2}})]_{y}^{1}

带入上下限计算后得:

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

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

化简

= \frac{e ^{y}}{y} - \frac{2 e ^{y}}{y ^{2}} + \frac{2 e ^{y} + 2 ( y ^{2} e ^{y^{2}} - e ^{y^{2}} )}{y ^{3}} - y e^{y^{2}}

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