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人気のある微分積分 >

integral of (2x)/((x-1)^2)

解

\int \frac{2 x}{( x - 1 ) ^{2}} d x

解

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

解答ステップ

\int \frac{2 x}{( x - 1 ) ^{2}} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{x}{( x - 1 ) ^{2}} d x

u置換積分法を適用する

= 2 \cdot \int \frac{u + 1}{u ^{2}} d u

拡張

\frac{u + 1}{u ^{2}} : \frac{1}{u} + \frac{1}{u ^{2}}

= 2 \cdot \int \frac{1}{u} + \frac{1}{u ^{2}} d u

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

= 2 (\int \frac{1}{u} d u + \int \frac{1}{u ^{2}} d u)

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{1}{u ^{2}} d u = - \frac{1}{u}

= 2 (ln ∣ u ∣ - \frac{1}{u})

代用を戻す

u = x - 1

= 2 (ln ∣ x - 1 ∣ - \frac{1}{x - 1})

定数を解答に追加する

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

グラフ

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