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

解答

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

\frac{x ^{3}}{9} + \frac{x ^{2}}{6} + \frac{1}{3} x - \frac{1}{3} ln ∣ 3 x - 3 ∣ + C

求解步骤

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

长除

\frac{x ^{3} - 2}{3 x - 3} : \frac{x ^{2}}{3} + \frac{x}{3} + \frac{1}{3} - \frac{1}{3 x - 3}

= \int \frac{x ^{2}}{3} + \frac{x}{3} + \frac{1}{3} - \frac{1}{3 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 \frac{x ^{2}}{3} d x + \int \frac{x}{3} d x + \int \frac{1}{3} d x - \int \frac{1}{3 x - 3} d x

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

\int \frac{x}{3} d x = \frac{x ^{2}}{6}

\int \frac{1}{3} d x = \frac{1}{3} x

\int \frac{1}{3 x - 3} d x = \frac{1}{3} ln ∣ 3 x - 3 ∣

= \frac{x ^{3}}{9} + \frac{x ^{2}}{6} + \frac{1}{3} x - \frac{1}{3} ln ∣ 3 x - 3 ∣

解答补常数

= \frac{x ^{3}}{9} + \frac{x ^{2}}{6} + \frac{1}{3} x - \frac{1}{3} ln ∣ 3 x - 3 ∣ + C

\int \frac{1}{( x ^{2} + 9 ) ^{3}} d x

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