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人気のある微分積分 >

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

解

\int \frac{x ^{2} + 2 x + 2}{x + 2} d x

解

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

解答ステップ

\int \frac{x ^{2} + 2 x + 2}{x + 2} d x

u置換積分法を適用する

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

拡張

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

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

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

\int 2 d u = 2 u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= \frac{u ^{2}}{2} - 2 u + 2 ln ∣ u ∣

代用を戻す

u = x + 2

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

定数を解答に追加する

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

グラフ

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