integral of (3x^2-10x-8)/(x-3)

解

\int \frac{3 x ^{2} - 10 x - 8}{x - 3} d x

8 x + \frac{3}{2} (x - 3)^{2} - 11 ln ∣ x - 3 ∣ - 24 + C

解答ステップ

\int \frac{3 x ^{2} - 10 x - 8}{x - 3} d x

拡張

\frac{3 x ^{2} - 10 x - 8}{x - 3} : \frac{3 x ^{2}}{x - 3} - \frac{10 x}{x - 3} - \frac{8}{x - 3}

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{3 x ^{2}}{x - 3} d x - \int \frac{10 x}{x - 3} d x - \int \frac{8}{x - 3} d x

\int \frac{3 x ^{2}}{x - 3} d x = 3 (\frac{( x - 3 ) ^{2}}{2} + 6 (x - 3) + 9 ln ∣ x - 3 ∣)

\int \frac{10 x}{x - 3} d x = 10 (x - 3 + 3 ln ∣ x - 3 ∣)

\int \frac{8}{x - 3} d x = 8 ln ∣ x - 3 ∣

= 3 (\frac{( x - 3 ) ^{2}}{2} + 6 (x - 3) + 9 ln ∣ x - 3 ∣) - 10 (x - 3 + 3 ln ∣ x - 3 ∣) - 8 ln ∣ x - 3 ∣

簡素化

3 (\frac{( x - 3 ) ^{2}}{2} + 6 (x - 3) + 9 ln ∣ x - 3 ∣) - 10 (x - 3 + 3 ln ∣ x - 3 ∣) - 8 ln ∣ x - 3 ∣ : 8 x + \frac{3}{2} (x - 3)^{2} - 11 ln ∣ x - 3 ∣ - 24

= 8 x + \frac{3}{2} (x - 3)^{2} - 11 ln ∣ x - 3 ∣ - 24

定数を解答に追加する

= 8 x + \frac{3}{2} (x - 3)^{2} - 11 ln ∣ x - 3 ∣ - 24 + C