integral of (ln(x^3))^2

解

\int (ln (x^{3}))^{2} d x

9 x ln^{2} (x) - 6 (3 x ln (x) - 3 x) + C

解答ステップ

\int (ln (x^{3}))^{2} d x

簡素化

= \int ln^{2} (x^{3}) d x

部分積分を適用する

= x ln^{2} (x^{3}) - \int 6 ln (x^{3}) d x

\int 6 ln (x^{3}) d x = 6 (3 x ln (x) - 3 x)

= x ln^{2} (x^{3}) - 6 (3 x ln (x) - 3 x)

簡素化

x ln^{2} (x^{3}) - 6 (3 x ln (x) - 3 x) : 9 x ln^{2} (x) - 6 (3 x ln (x) - 3 x)

= 9 x ln^{2} (x) - 6 (3 x ln (x) - 3 x)

定数を解答に追加する

= 9 x ln^{2} (x) - 6 (3 x ln (x) - 3 x) + C