integral of (x^2-5x-24)/(x+3)

解

\int \frac{x ^{2} - 5 x - 24}{x + 3} d x

- 11 x + \frac{1}{2} (x + 3)^{2} - 33 + C

解答ステップ

\int \frac{x ^{2} - 5 x - 24}{x + 3} d x

拡張

\frac{x ^{2} - 5 x - 24}{x + 3} : \frac{x ^{2}}{x + 3} - \frac{5 x}{x + 3} - \frac{24}{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{x ^{2}}{x + 3} d x - \int \frac{5 x}{x + 3} d x - \int \frac{24}{x + 3} d x

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

\int \frac{5 x}{x + 3} d x = 5 (x + 3 - 3 ln ∣ x + 3 ∣)

\int \frac{24}{x + 3} d x = 24 ln ∣ x + 3 ∣

= \frac{( x + 3 ) ^{2}}{2} - 6 (x + 3) + 9 ln ∣ x + 3 ∣ - 5 (x + 3 - 3 ln ∣ x + 3 ∣) - 24 ln ∣ x + 3 ∣

簡素化

\frac{( x + 3 ) ^{2}}{2} - 6 (x + 3) + 9 ln ∣ x + 3 ∣ - 5 (x + 3 - 3 ln ∣ x + 3 ∣) - 24 ln ∣ x + 3 ∣ : - 11 x + \frac{1}{2} (x + 3)^{2} - 33

= - 11 x + \frac{1}{2} (x + 3)^{2} - 33

定数を解答に追加する

= - 11 x + \frac{1}{2} (x + 3)^{2} - 33 + C