integral of ((2x))/((x-2)(x+1))

解

\int \frac{( 2 x )}{( x - 2 ) ( x + 1 )} d x

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

解答ステップ

\int \frac{2 x}{( x - 2 ) ( x + 1 )} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{x}{( x - 2 ) ( x + 1 )} d x

以下の部分分数を得る:

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

= 2 \cdot \int \frac{2}{3 ( x - 2 )} + \frac{1}{3 ( x + 1 )} 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

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

\int \frac{2}{3 ( x - 2 )} d x = \frac{2}{3} ln ∣ x - 2 ∣

\int \frac{1}{3 ( x + 1 )} d x = \frac{1}{3} ln ∣ x + 1 ∣

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

定数を解答に追加する

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