integral of (2x^3+4x^2-5)/(x+3)

解

\int \frac{2 x ^{3} + 4 x ^{2} - 5}{x + 3} d x

\frac{2 x ^{3}}{3} - x^{2} + 6 x - 23 ln ∣ x + 3 ∣ - 54 + C

解答ステップ

\int \frac{2 x ^{3} + 4 x ^{2} - 5}{x + 3} d x

拡張

\frac{2 x ^{3} + 4 x ^{2} - 5}{x + 3} : \frac{2 x ^{3}}{x + 3} + \frac{4 x ^{2}}{x + 3} - \frac{5}{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 ^{3}}{x + 3} d x + \int \frac{4 x ^{2}}{x + 3} d x - \int \frac{5}{x + 3} d x

\int \frac{2 x ^{3}}{x + 3} d x = \frac{2 x ^{3}}{3} - 3 x^{2} + 18 x - 54 ln ∣ x + 3 ∣

\int \frac{4 x ^{2}}{x + 3} d x = 4 (\frac{( x + 3 ) ^{2}}{2} - 6 (x + 3) + 9 ln ∣ x + 3 ∣)

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

= \frac{2 x ^{3}}{3} - 3 x^{2} + 18 x - 54 ln ∣ x + 3 ∣ + 4 (\frac{( x + 3 ) ^{2}}{2} - 6 (x + 3) + 9 ln ∣ x + 3 ∣) - 5 ln ∣ x + 3 ∣

簡素化

\frac{2 x ^{3}}{3} - 3 x^{2} + 18 x - 54 ln ∣ x + 3 ∣ + 4 (\frac{( x + 3 ) ^{2}}{2} - 6 (x + 3) + 9 ln ∣ x + 3 ∣) - 5 ln ∣ x + 3 ∣ : \frac{2 x ^{3}}{3} - x^{2} + 6 x - 23 ln ∣ x + 3 ∣ - 54

= \frac{2 x ^{3}}{3} - x^{2} + 6 x - 23 ln ∣ x + 3 ∣ - 54

定数を解答に追加する

= \frac{2 x ^{3}}{3} - x^{2} + 6 x - 23 ln ∣ x + 3 ∣ - 54 + C