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

解

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

\frac{x ^{2}}{2} + x - ln 4 x^{2} - 4 x + 12 - \frac{12}{11} arctan (\frac{1}{11} (2 x - 1)) + C

解答ステップ

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

長除法

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

= \int x + 1 + \frac{- 2 x - 5}{x ^{2} - x + 3} 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 x d x + \int 1 d x + \int \frac{- 2 x - 5}{x ^{2} - x + 3} d x

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

\int 1 d x = x

\int \frac{- 2 x - 5}{x ^{2} - x + 3} d x = - ln 4 x^{2} - 4 x + 12 - \frac{12}{11} arctan (\frac{1}{11} (2 x - 1))

= \frac{x ^{2}}{2} + x - ln 4 x^{2} - 4 x + 12 - \frac{12}{11} arctan (\frac{1}{11} (2 x - 1))

定数を解答に追加する

= \frac{x ^{2}}{2} + x - ln 4 x^{2} - 4 x + 12 - \frac{12}{11} arctan (\frac{1}{11} (2 x - 1)) + C