integral from 0 to 1/13 of xln(13x)

解答

\int_{0}^{\frac{1}{13}} x ln (13 x) d x

- \frac{1}{676}

十进制

- 0.00147 \dots

求解步骤

\int_{0}^{\frac{1}{13}} x ln (13 x) d x

使用分布积分法

= [\frac{1}{2} (x^{2} ln (13 x) - 2 \cdot \int \frac{x}{2} d x)]_{0}^{\frac{1}{13}}

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

= [\frac{1}{2} (x^{2} ln (13 x) - 2 \cdot \frac{x ^{2}}{4})]_{0}^{\frac{1}{13}}

2 \cdot \frac{x ^{2}}{4} = \frac{x ^{2}}{2}

= [\frac{1}{2} (x^{2} ln (13 x) - \frac{x ^{2}}{2})]_{0}^{\frac{1}{13}}

带入上下限计算后得:

- \frac{1}{676}

= - \frac{1}{676}

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