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

解

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

ln ∣ x - 2 ∣ - \frac{4}{x - 2} + C

解答ステップ

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

拡張

\frac{x + 2}{( x - 2 ) ^{2}} : \frac{x}{( x - 2 ) ^{2}} + \frac{2}{( x - 2 ) ^{2}}

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

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

\int \frac{2}{( x - 2 ) ^{2}} d x = - \frac{2}{x - 2}

= ln ∣ x - 2 ∣ - \frac{2}{x - 2} - \frac{2}{x - 2}

簡素化

ln ∣ x - 2 ∣ - \frac{2}{x - 2} - \frac{2}{x - 2} : ln ∣ x - 2 ∣ - \frac{4}{x - 2}

= ln ∣ x - 2 ∣ - \frac{4}{x - 2}

定数を解答に追加する

= ln ∣ x - 2 ∣ - \frac{4}{x - 2} + C