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

解

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

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

解答ステップ

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

拡張

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

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

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

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

定数を解答に追加する

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