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

解

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

\frac{x ^{2}}{4} - \frac{1}{2} x + ln ∣ x - 1 ∣ + 2 ln ∣ x + 2 ∣ + \frac{1}{3} (ln \frac{4}{3} + \frac{2}{3} x - ln - \frac{2}{3} + \frac{2}{3} x) + C

解答ステップ

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

長除法

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

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

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

\int \frac{1}{2} d x = \frac{1}{2} x

\int \frac{3 x - 1}{x ^{2} + x - 2} d x = ln ∣ x - 1 ∣ + 2 ln ∣ x + 2 ∣ + \frac{1}{3} (ln \frac{4}{3} + \frac{2}{3} x - ln - \frac{2}{3} + \frac{2}{3} x)

= \frac{x ^{2}}{4} - \frac{1}{2} x + ln ∣ x - 1 ∣ + 2 ln ∣ x + 2 ∣ + \frac{1}{3} (ln \frac{4}{3} + \frac{2}{3} x - ln - \frac{2}{3} + \frac{2}{3} x)

定数を解答に追加する

= \frac{x ^{2}}{4} - \frac{1}{2} x + ln ∣ x - 1 ∣ + 2 ln ∣ x + 2 ∣ + \frac{1}{3} (ln \frac{4}{3} + \frac{2}{3} x - ln - \frac{2}{3} + \frac{2}{3} x) + C