integral of 1/(x(2000-x))

解

\int \frac{1}{x ( 2000 - x )} d x

\frac{1}{2000} ln ∣ x ∣ - \frac{1}{2000} ln ∣ x - 2000 ∣ + C

解答ステップ

\int \frac{1}{x ( 2000 - x )} d x

以下の部分分数を得る:

\frac{1}{x ( 2000 - x )} : \frac{1}{2000 x} - \frac{1}{2000 ( x - 2000 )}

= \int \frac{1}{2000 x} - \frac{1}{2000 ( x - 2000 )} 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}{2000 x} d x - \int \frac{1}{2000 ( x - 2000 )} d x

\int \frac{1}{2000 x} d x = \frac{1}{2000} ln ∣ x ∣

\int \frac{1}{2000 ( x - 2000 )} d x = \frac{1}{2000} ln ∣ x - 2000 ∣

= \frac{1}{2000} ln ∣ x ∣ - \frac{1}{2000} ln ∣ x - 2000 ∣

定数を解答に追加する

= \frac{1}{2000} ln ∣ x ∣ - \frac{1}{2000} ln ∣ x - 2000 ∣ + C