integral from 1 to e of yln(y)

解

\int_{1}^{e} y ln (y) d y

\frac{e ^{2} + 1}{4}

十進法表記

2.09726 \dots

解答ステップ

\int_{1}^{e} y ln (y) d y

部分積分を適用する

= [\frac{1}{2} (y^{2} ln (y) - 2 \cdot \int \frac{y}{2} d y)]_{1}^{e}

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

= [\frac{1}{2} (y^{2} ln (y) - 2 \cdot \frac{y ^{2}}{4})]_{1}^{e}

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

= [\frac{1}{2} (y^{2} ln (y) - \frac{y ^{2}}{2})]_{1}^{e}

限度を計算する:

\frac{e ^{2}}{4} + \frac{1}{4}

= \frac{e ^{2}}{4} + \frac{1}{4}

簡素化

= \frac{e ^{2} + 1}{4}