integral from 1/6 to e of |ln(x)|

解答

\int_{\frac{1}{6}}^{e} ∣ ln (x) ∣ d x

\frac{11}{6} - \frac{1}{6} ln (6)

十进制

1.53470 \dots

求解步骤

\int_{\frac{1}{6}}^{e} ∣ ln (x) ∣ d x

使用换元积分法

= \int_{- l n (6)}^{1} ∣ u ∣ e^{u} d u

拿掉绝对值

= \int_{- l n (6)}^{0} (- u) e^{u} d u + \int_{0}^{1} u e^{u} d u

\int_{- l n (6)}^{0} (- u) e^{u} d u = \frac{5}{6} - \frac{1}{6} ln (6)

\int_{0}^{1} u e^{u} d u = 1

= \frac{5}{6} - \frac{1}{6} ln (6) + 1

合并分式

1 + \frac{5}{6} : \frac{11}{6}

= \frac{11}{6} - \frac{1}{6} ln (6)

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