integral of (x^3-x^2-2x)/(4x-2)

解

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

\frac{x ^{3}}{12} - \frac{x ^{2}}{16} - \frac{9}{16} x - \frac{9}{32} ln ∣ 4 x - 2 ∣ + C

解答ステップ

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

長除法

\frac{x ^{3} - x ^{2} - 2 x}{4 x - 2} : \frac{x ^{2}}{4} - \frac{x}{8} - \frac{9}{16} - \frac{9}{8 ( 4 x - 2 )}

= \int \frac{x ^{2}}{4} - \frac{x}{8} - \frac{9}{16} - \frac{9}{8 ( 4 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 \frac{x ^{2}}{4} d x - \int \frac{x}{8} d x - \int \frac{9}{16} d x - \int \frac{9}{8 ( 4 x - 2 )} d x

\int \frac{x ^{2}}{4} d x = \frac{x ^{3}}{12}

\int \frac{x}{8} d x = \frac{x ^{2}}{16}

\int \frac{9}{16} d x = \frac{9}{16} x

\int \frac{9}{8 ( 4 x - 2 )} d x = \frac{9}{32} ln ∣ 4 x - 2 ∣

= \frac{x ^{3}}{12} - \frac{x ^{2}}{16} - \frac{9}{16} x - \frac{9}{32} ln ∣ 4 x - 2 ∣

定数を解答に追加する

= \frac{x ^{3}}{12} - \frac{x ^{2}}{16} - \frac{9}{16} x - \frac{9}{32} ln ∣ 4 x - 2 ∣ + C