integral of (ln(y))^2y

解

\int (ln (y))^{2} y d y

\frac{1}{2} y^{2} ln^{2} (y) - \frac{1}{2} y^{2} ln (y) + \frac{y ^{2}}{4} + C

解答ステップ

\int (ln (y))^{2} y d y

部分積分を適用する

= \frac{1}{2} y^{2} ln^{2} (y) - \int y ln (y) d y

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

= \frac{1}{2} y^{2} ln^{2} (y) - (\frac{1}{2} y^{2} ln (y) - \frac{y ^{2}}{4})

簡素化

= \frac{1}{2} y^{2} ln^{2} (y) - \frac{1}{2} y^{2} ln (y) + \frac{y ^{2}}{4}

定数を解答に追加する

= \frac{1}{2} y^{2} ln^{2} (y) - \frac{1}{2} y^{2} ln (y) + \frac{y ^{2}}{4} + C