integral of (x^3+3x^2-9x-2)/(x-2)

解答

\int \frac{x ^{3} + 3 x ^{2} - 9 x - 2}{x - 2} d x

3 x^{2} + 3 x + \frac{1}{3} (x - 2)^{3} + \frac{3}{2} (x - 2)^{2} - 18 + C

求解步骤

\int \frac{x ^{3} + 3 x ^{2} - 9 x - 2}{x - 2} d x

乘开

\frac{x ^{3} + 3 x ^{2} - 9 x - 2}{x - 2} : \frac{x ^{3}}{x - 2} + \frac{3 x ^{2}}{x - 2} - \frac{9 x}{x - 2} - \frac{2}{x - 2}

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x ^{3}}{x - 2} d x + \int \frac{3 x ^{2}}{x - 2} d x - \int \frac{9 x}{x - 2} d x - \int \frac{2}{x - 2} d x

\int \frac{x ^{3}}{x - 2} d x = 3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12

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

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

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

= 3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12 + 3 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣) - 9 (x - 2 + 2 ln ∣ x - 2 ∣) - 2 ln ∣ x - 2 ∣

化简

3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12 + 3 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣) - 9 (x - 2 + 2 ln ∣ x - 2 ∣) - 2 ln ∣ x - 2 ∣ : 3 x^{2} + 3 x + \frac{1}{3} (x - 2)^{3} + \frac{3}{2} (x - 2)^{2} - 18

= 3 x^{2} + 3 x + \frac{1}{3} (x - 2)^{3} + \frac{3}{2} (x - 2)^{2} - 18

解答补常数

= 3 x^{2} + 3 x + \frac{1}{3} (x - 2)^{3} + \frac{3}{2} (x - 2)^{2} - 18 + C