integral of (x^2-x+12)/(x^3+2x)

解

\int \frac{x ^{2} - x + 12}{x ^{3} + 2 x} d x

6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 2 - \frac{1}{2} arctan (\frac{x}{2}) + C

解答ステップ

\int \frac{x ^{2} - x + 12}{x ^{3} + 2 x} d x

以下の部分分数を得る:

\frac{x ^{2} - x + 12}{x ^{3} + 2 x} : \frac{6}{x} + \frac{- 5 x - 1}{x ^{2} + 2}

= \int \frac{6}{x} + \frac{- 5 x - 1}{x ^{2} + 2} d 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{6}{x} d x + \int \frac{- 5 x - 1}{x ^{2} + 2} d x

\int \frac{6}{x} d x = 6 ln ∣ x ∣

\int \frac{- 5 x - 1}{x ^{2} + 2} d x = - \frac{5}{2} ln x^{2} + 2 - \frac{1}{2} arctan (\frac{x}{2})

= 6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 2 - \frac{1}{2} arctan (\frac{x}{2})

定数を解答に追加する

= 6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 2 - \frac{1}{2} arctan (\frac{x}{2}) + C