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

解

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

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

解答ステップ

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

拡張

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

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

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

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

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

簡素化

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

定数を解答に追加する

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