integral of ((ln(x))^2)/(4x^3)

解

\int \frac{( ln ( x ) ) ^{2}}{4 x ^{3}} d x

\frac{- 2 ln ^{2} ( x ) - 2 ln ( x ) - 1}{16 x ^{2}} + C

解答ステップ

\int \frac{( ln ( x ) ) ^{2}}{4 x ^{3}} d x

定数を除く:

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

= \frac{1}{4} \cdot \int \frac{( ln ( x ) ) ^{2}}{x ^{3}} d x

部分積分を適用する

= \frac{1}{4} (- \frac{ln ^{2} ( x )}{2 x ^{2}} - \int - \frac{ln ( x )}{x ^{3}} d x)

\int - \frac{ln ( x )}{x ^{3}} d x = \frac{ln ( x )}{2 x ^{2}} + \frac{1}{4 x ^{2}}

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

簡素化

\frac{1}{4} (- \frac{ln ^{2} ( x )}{2 x ^{2}} - (\frac{ln ( x )}{2 x ^{2}} + \frac{1}{4 x ^{2}})) : \frac{- 2 ln ^{2} ( x ) - 2 ln ( x ) - 1}{16 x ^{2}}

= \frac{- 2 ln ^{2} ( x ) - 2 ln ( x ) - 1}{16 x ^{2}}

定数を解答に追加する

= \frac{- 2 ln ^{2} ( x ) - 2 ln ( x ) - 1}{16 x ^{2}} + C