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

解

\int_{\frac{1}{6}}^{e} ∣ ln (x) ∣ d x

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

十進法表記

1.53470 \dots

解答ステップ

\int_{\frac{1}{6}}^{e} ∣ ln (x) ∣ d x

u置換積分法を適用する

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

絶対数を排除する

= \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

分数を組み合わせる

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

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