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

解

\int \frac{x ^{3}}{x ^{3} - x} d x

\frac{x ^{3}}{x ^{2} - 1} + 2 (- \frac{1}{4} ln ∣ x + 1 ∣ - \frac{1}{4 ( x + 1 )} + \frac{1}{4} ln ∣ x - 1 ∣ - \frac{1}{4 ( x - 1 )}) + C

解答ステップ

\int \frac{x ^{3}}{x ^{3} - x} d x

部分積分を適用する

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

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

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

簡素化

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

定数を解答に追加する

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