integral of x^4(ln(x))^2

解

\int x^{4} (ln (x))^{2} d x

\frac{1}{5} x^{5} ln^{2} (x) - \frac{2}{5} (\frac{1}{5} x^{5} ln (x) - \frac{x ^{5}}{25}) + C

解答ステップ

\int x^{4} (ln (x))^{2} d x

部分積分を適用する

= \frac{1}{5} x^{5} ln^{2} (x) - \int \frac{2}{5} x^{4} ln (x) d x

\int \frac{2}{5} x^{4} ln (x) d x = \frac{2}{5} (\frac{1}{5} x^{5} ln (x) - \frac{x ^{5}}{25})

= \frac{1}{5} x^{5} ln^{2} (x) - \frac{2}{5} (\frac{1}{5} x^{5} ln (x) - \frac{x ^{5}}{25})

定数を解答に追加する

= \frac{1}{5} x^{5} ln^{2} (x) - \frac{2}{5} (\frac{1}{5} x^{5} ln (x) - \frac{x ^{5}}{25}) + C