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人気のある微分積分 >

integral of (x^2+1)/((x-1)^3)

解

\int \frac{x ^{2} + 1}{( x - 1 ) ^{3}} d x

解

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

解答ステップ

\int \frac{x ^{2} + 1}{( x - 1 ) ^{3}} d x

u置換積分法を適用する

= \int \frac{u ^{2} + 2 u + 2}{u ^{3}} d u

拡張

\frac{u ^{2} + 2 u + 2}{u ^{3}} : \frac{1}{u} + \frac{2}{u ^{2}} + \frac{2}{u ^{3}}

= \int \frac{1}{u} + \frac{2}{u ^{2}} + \frac{2}{u ^{3}} 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

= \int \frac{1}{u} d u + \int \frac{2}{u ^{2}} d u + \int \frac{2}{u ^{3}} d u

\int \frac{1}{u} d u = ln ∣ u ∣

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

\int \frac{2}{u ^{3}} d u = - \frac{1}{u ^{2}}

= ln ∣ u ∣ - \frac{2}{u} - \frac{1}{u ^{2}}

代用を戻す

u = x - 1

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

定数を解答に追加する

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

グラフ

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