積分記号 (x^3)/((x^2+x+2))

解

積分記号

\frac{x ^{3}}{( x ^{2} + x + 2 )}

\frac{x ^{2}}{2} - x - \frac{1}{2} ln 4 x^{2} + 4 x + 8 + \frac{5}{7} arctan (\frac{2 x + 1}{7}) + C

解答ステップ

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

長除法

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

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

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

\int 1 d x = x

\int \frac{- x + 2}{x ^{2} + x + 2} d x = - \frac{1}{2} ln 4 x^{2} + 4 x + 8 + \frac{5}{7} arctan (\frac{2 x + 1}{7})

= \frac{x ^{2}}{2} - x - \frac{1}{2} ln 4 x^{2} + 4 x + 8 + \frac{5}{7} arctan (\frac{2 x + 1}{7})

定数を解答に追加する

= \frac{x ^{2}}{2} - x - \frac{1}{2} ln 4 x^{2} + 4 x + 8 + \frac{5}{7} arctan (\frac{2 x + 1}{7}) + C