integral of (x^2+2x-2)/(x^3-x^2-2x)

解

\int \frac{x ^{2} + 2 x - 2}{x ^{3} - x ^{2} - 2 x} d x

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

解答ステップ

\int \frac{x ^{2} + 2 x - 2}{x ^{3} - x ^{2} - 2 x} d x

拡張

\frac{x ^{2} + 2 x - 2}{x ^{3} - x ^{2} - 2 x} : \frac{x ^{2}}{x ^{3} - x ^{2} - 2 x} + \frac{2 x}{x ^{3} - x ^{2} - 2 x} - \frac{2}{x ^{3} - x ^{2} - 2 x}

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x ^{2}}{x ^{3} - x ^{2} - 2 x} d x + \int \frac{2 x}{x ^{3} - x ^{2} - 2 x} d x - \int \frac{2}{x ^{3} - x ^{2} - 2 x} d x

\int \frac{x ^{2}}{x ^{3} - x ^{2} - 2 x} d x = \frac{1}{3} ln ∣ x + 1 ∣ + \frac{2}{3} ln ∣ x - 2 ∣

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

\int \frac{2}{x ^{3} - x ^{2} - 2 x} d x = - ln ∣ x ∣ + \frac{2}{3} ln ∣ x + 1 ∣ + \frac{1}{3} ln ∣ x - 2 ∣

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

簡素化

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

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

定数を解答に追加する

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