integral from 1 to 3 of (x-1)ln(x)

解

\int_{1}^{3} (x - 1) ln (x) d x

\frac{3}{2} ln (3)

十進法表記

1.64791 \dots

解答ステップ

\int_{1}^{3} (x - 1) ln (x) d x

部分積分を適用する

= [ln (x) (\frac{x ^{2}}{2} - x) - \int \frac{1}{2} (x - 2) d x]_{1}^{3}

\int \frac{1}{2} (x - 2) d x = \frac{x ^{2}}{4} - x

= [ln (x) (\frac{x ^{2}}{2} - x) - (\frac{x ^{2}}{4} - x)]_{1}^{3}

簡素化

[ln (x) (\frac{x ^{2}}{2} - x) - (\frac{x ^{2}}{4} - x)]_{1}^{3} : [\frac{1}{2} x^{2} ln (x) - x ln (x) - \frac{x ^{2}}{4} + x]_{1}^{3}

= [\frac{1}{2} x^{2} ln (x) - x ln (x) - \frac{x ^{2}}{4} + x]_{1}^{3}

限度を計算する:

\frac{3}{2} ln (3)

= \frac{3}{2} ln (3)