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

解答

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

\frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} + 9 x - 29 ln ∣ x + 3 ∣ + C

求解步骤

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

长除

\frac{x ^{3} - 2}{x + 3} : x^{2} - 3 x + 9 - \frac{29}{x + 3}

= \int x^{2} - 3 x + 9 - \frac{29}{x + 3} d x

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 x^{2} d x - \int 3 x d x + \int 9 d x - \int \frac{29}{x + 3} d x

\int x^{2} d x = \frac{x ^{3}}{3}

\int 3 x d x = \frac{3 x ^{2}}{2}

\int 9 d x = 9 x

\int \frac{29}{x + 3} d x = 29 ln ∣ x + 3 ∣

= \frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} + 9 x - 29 ln ∣ x + 3 ∣

解答补常数

= \frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} + 9 x - 29 ln ∣ x + 3 ∣ + C

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