integral of (2x^2-4x+3)/(x-2)

解

\int \frac{2 x ^{2} - 4 x + 3}{x - 2} d x

x^{2} + 3 ln ∣ x - 2 ∣ - 4 + C

解答ステップ

\int \frac{2 x ^{2} - 4 x + 3}{x - 2} d x

拡張

\frac{2 x ^{2} - 4 x + 3}{x - 2} : \frac{2 x ^{2}}{x - 2} - \frac{4 x}{x - 2} + \frac{3}{x - 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{2 x ^{2}}{x - 2} d x - \int \frac{4 x}{x - 2} d x + \int \frac{3}{x - 2} d x

\int \frac{2 x ^{2}}{x - 2} d x = x^{2} + 4 x + 8 ln ∣ x - 2 ∣ - 12

\int \frac{4 x}{x - 2} d x = 4 (x - 2 + 2 ln ∣ x - 2 ∣)

\int \frac{3}{x - 2} d x = 3 ln ∣ x - 2 ∣

= x^{2} + 4 x + 8 ln ∣ x - 2 ∣ - 12 - 4 (x - 2 + 2 ln ∣ x - 2 ∣) + 3 ln ∣ x - 2 ∣

簡素化

x^{2} + 4 x + 8 ln ∣ x - 2 ∣ - 12 - 4 (x - 2 + 2 ln ∣ x - 2 ∣) + 3 ln ∣ x - 2 ∣ : x^{2} + 3 ln ∣ x - 2 ∣ - 4

= x^{2} + 3 ln ∣ x - 2 ∣ - 4

定数を解答に追加する

= x^{2} + 3 ln ∣ x - 2 ∣ - 4 + C