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

解

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

\frac{1}{5} ln ∣ x - 2 ∣ - \frac{1}{5} ln ∣ x + 3 ∣ + C

解答ステップ

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

共通因数を約分する:

x

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

以下の部分分数を得る:

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

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

\int \frac{1}{5 ( x - 2 )} d x = \frac{1}{5} ln ∣ x - 2 ∣

\int \frac{1}{5 ( x + 3 )} d x = \frac{1}{5} ln ∣ x + 3 ∣

= \frac{1}{5} ln ∣ x - 2 ∣ - \frac{1}{5} ln ∣ x + 3 ∣

定数を解答に追加する

= \frac{1}{5} ln ∣ x - 2 ∣ - \frac{1}{5} ln ∣ x + 3 ∣ + C