integral of 8x(ln(x^5))^2

解

\int 8 x (ln (x^{5}))^{2} d x

8 (\frac{25}{2} x^{2} ln^{2} (x) - 5 (\frac{5}{2} x^{2} ln (x) - \frac{5 x ^{2}}{4})) + C

解答ステップ

\int 8 x (ln (x^{5}))^{2} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 8 \cdot \int x (ln (x^{5}))^{2} d x

簡素化

= 8 \cdot \int x ln^{2} (x^{5}) d x

部分積分を適用する

= 8 (\frac{1}{2} x^{2} ln^{2} (x^{5}) - \int 5 x ln (x^{5}) d x)

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

= 8 (\frac{1}{2} x^{2} ln^{2} (x^{5}) - 5 (\frac{5}{2} x^{2} ln (x) - \frac{5 x ^{2}}{4}))

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

= 8 (\frac{25}{2} x^{2} ln^{2} (x) - 5 (\frac{5}{2} x^{2} ln (x) - \frac{5 x ^{2}}{4}))

定数を解答に追加する

= 8 (\frac{25}{2} x^{2} ln^{2} (x) - 5 (\frac{5}{2} x^{2} ln (x) - \frac{5 x ^{2}}{4})) + C