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

解

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

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

解答ステップ

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

拡張

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

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

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

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

簡素化

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

定数を解答に追加する

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