integral of (-5x^2+11x-10)/(x^3-x^2-6x)

解

\int \frac{- 5 x ^{2} + 11 x - 10}{x ^{3} - x ^{2} - 6 x} d x

\frac{5}{3} ln ∣ x ∣ - \frac{22}{15} ln ∣ x - 3 ∣ - \frac{26}{5} ln ∣ x + 2 ∣ + C

解答ステップ

\int \frac{- 5 x ^{2} + 11 x - 10}{x ^{3} - x ^{2} - 6 x} d x

拡張

\frac{- 5 x ^{2} + 11 x - 10}{x ^{3} - x ^{2} - 6 x} : - \frac{5 x ^{2}}{x ^{3} - x ^{2} - 6 x} + \frac{11 x}{x ^{3} - x ^{2} - 6 x} - \frac{10}{x ^{3} - x ^{2} - 6 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{5 x ^{2}}{x ^{3} - x ^{2} - 6 x} d x + \int \frac{11 x}{x ^{3} - x ^{2} - 6 x} d x - \int \frac{10}{x ^{3} - x ^{2} - 6 x} d x

\int \frac{5 x ^{2}}{x ^{3} - x ^{2} - 6 x} d x = 2 ln ∣ x + 2 ∣ + 3 ln ∣ x - 3 ∣

\int \frac{11 x}{x ^{3} - x ^{2} - 6 x} d x = 11 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣)

\int \frac{10}{x ^{3} - x ^{2} - 6 x} d x = - \frac{5}{3} ln ∣ x ∣ + ln ∣ x + 2 ∣ + \frac{2}{3} ln ∣ x - 3 ∣

= - (2 ln ∣ x + 2 ∣ + 3 ln ∣ x - 3 ∣) + 11 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣) - (- \frac{5}{3} ln ∣ x ∣ + ln ∣ x + 2 ∣ + \frac{2}{3} ln ∣ x - 3 ∣)

簡素化

- (2 ln ∣ x + 2 ∣ + 3 ln ∣ x - 3 ∣) + 11 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣) - (- \frac{5}{3} ln ∣ x ∣ + ln ∣ x + 2 ∣ + \frac{2}{3} ln ∣ x - 3 ∣) : \frac{5}{3} ln ∣ x ∣ - \frac{22}{15} ln ∣ x - 3 ∣ - \frac{26}{5} ln ∣ x + 2 ∣

= \frac{5}{3} ln ∣ x ∣ - \frac{22}{15} ln ∣ x - 3 ∣ - \frac{26}{5} ln ∣ x + 2 ∣

定数を解答に追加する

= \frac{5}{3} ln ∣ x ∣ - \frac{22}{15} ln ∣ x - 3 ∣ - \frac{26}{5} ln ∣ x + 2 ∣ + C