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

解答

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

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

求解步骤

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

长除

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

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

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

\int \frac{x}{3 x ^{2} + 3} d x = \frac{1}{6} ln 3 x^{2} + 3

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

解答补常数

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

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