integral from 0.1 to 2/pi of x^2sin(1/x)

解答

\int_{0.1}^{\frac{2}{π}} x^{2} sin (\frac{1}{x}) d x

\frac{31.00627 \dots Ci ( \frac{1}{\frac{2}{π}} ) - 5.13211 \dots - 31.00627 \dots Ci ( \frac{1}{0.1} )}{186.03766 \dots}

求解步骤

\int_{0.1}^{\frac{2}{π}} x^{2} sin (\frac{1}{x}) d x

使用分布积分法

= [\frac{1}{3} (x^{3} sin (\frac{1}{x}) - 3 \cdot \int - \frac{x cos ( \frac{1}{x} )}{3} d x)]_{0.1}^{\frac{2}{π}}

\int - \frac{x cos ( \frac{1}{x} )}{3} d x = - \frac{1}{3} (\frac{1}{2} x^{2} cos (\frac{1}{x}) - \frac{1}{2} (x sin (\frac{1}{x}) - Ci (\frac{1}{x})))

= [\frac{1}{3} (x^{3} sin (\frac{1}{x}) - 3 (- \frac{1}{3} (\frac{1}{2} x^{2} cos (\frac{1}{x}) - \frac{1}{2} (x sin (\frac{1}{x}) - Ci (\frac{1}{x})))))]_{0.1}^{\frac{2}{π}}

化简

[\frac{1}{3} (x^{3} sin (\frac{1}{x}) - 3 (- \frac{1}{3} (\frac{1}{2} x^{2} cos (\frac{1}{x}) - \frac{1}{2} (x sin (\frac{1}{x}) - Ci (\frac{1}{x})))))]_{0.1}^{\frac{2}{π}} : [\frac{1}{6} (2 x^{3} sin (\frac{1}{x}) + x^{2} cos (\frac{1}{x}) - x sin (\frac{1}{x}) + Ci (\frac{1}{x}))]_{0.1}^{\frac{2}{π}}

= [\frac{1}{6} (2 x^{3} sin (\frac{1}{x}) + x^{2} cos (\frac{1}{x}) - x sin (\frac{1}{x}) + Ci (\frac{1}{x}))]_{0.1}^{\frac{2}{π}}

带入上下限计算后得:

\frac{31.00627 \dots Ci ( \frac{1}{\frac{2}{π}} ) - 5.13211 \dots - 31.00627 \dots Ci ( \frac{1}{0.1} )}{186.03766 \dots}

= \frac{31.00627 \dots Ci ( \frac{1}{\frac{2}{π}} ) - 5.13211 \dots - 31.00627 \dots Ci ( \frac{1}{0.1} )}{186.03766 \dots}

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