integral of x/(x+

解

\int \frac{x}{x + d} d x

- d ln ∣ x + d ∣ + x + C

解答ステップ

\int \frac{x}{x + d} d x

\frac{x}{x + d} = - \frac{d}{x + d} + 1

= \int - \frac{d}{x + d} + 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

= - \int \frac{d}{x + d} d x + \int 1 d x

\int \frac{d}{x + d} d x = d ln ∣ x + d ∣

\int 1 d x = x

= - d ln ∣ x + d ∣ + x

定数を解答に追加する

= - d ln ∣ x + d ∣ + x + C