integral of (2x^2+x+15)/(x+3)

解

\int \frac{2 x ^{2} + x + 15}{x + 3} d x

x^{2} - 5 x + 30 ln ∣ x + 3 ∣ - 24 + C

解答ステップ

\int \frac{2 x ^{2} + x + 15}{x + 3} d x

拡張

\frac{2 x ^{2} + x + 15}{x + 3} : \frac{2 x ^{2}}{x + 3} + \frac{x}{x + 3} + \frac{15}{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{2 x ^{2}}{x + 3} d x + \int \frac{x}{x + 3} d x + \int \frac{15}{x + 3} d x

\int \frac{2 x ^{2}}{x + 3} d x = x^{2} - 6 x + 18 ln ∣ x + 3 ∣ - 27

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

\int \frac{15}{x + 3} d x = 15 ln ∣ x + 3 ∣

= x^{2} - 6 x + 18 ln ∣ x + 3 ∣ - 27 + x + 3 - 3 ln ∣ x + 3 ∣ + 15 ln ∣ x + 3 ∣

簡素化

= x^{2} - 5 x + 30 ln ∣ x + 3 ∣ - 24

定数を解答に追加する

= x^{2} - 5 x + 30 ln ∣ x + 3 ∣ - 24 + C