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

解

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

- \frac{ln ^{3} ( x )}{2 x ^{2}} + \frac{3 ( - 2 ln ^{2} ( x ) - 2 ln ( x ) - 1 )}{8 x ^{2}} + C

解答ステップ

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

指数の規則を適用する:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

部分積分を適用する

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

\int - \frac{3 ln ^{2} ( x )}{2 x ^{3}} d x = - \frac{3 ( - 2 ln ^{2} ( x ) - 2 ln ( x ) - 1 )}{8 x ^{2}}

= - \frac{ln ^{3} ( x )}{2 x ^{2}} - (- \frac{3 ( - 2 ln ^{2} ( x ) - 2 ln ( x ) - 1 )}{8 x ^{2}})

簡素化

= - \frac{ln ^{3} ( x )}{2 x ^{2}} + \frac{3 ( - 2 ln ^{2} ( x ) - 2 ln ( x ) - 1 )}{8 x ^{2}}

定数を解答に追加する

= - \frac{ln ^{3} ( x )}{2 x ^{2}} + \frac{3 ( - 2 ln ^{2} ( x ) - 2 ln ( x ) - 1 )}{8 x ^{2}} + C