integral of (27x^3-2)/(\sqrt[3]{x)}

解答

\int \frac{27 x ^{3} - 2}{3 x} d x

\frac{81}{11} x^{\frac{11}{3}} - 3 x^{\frac{2}{3}} + C

求解步骤

\int \frac{27 x ^{3} - 2}{3 x} d x

乘开

\frac{27 x ^{3} - 2}{3 x} : 27 x^{\frac{8}{3}} - \frac{2}{3 x}

= \int 27 x^{\frac{8}{3}} - \frac{2}{3 x} 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 27 x^{\frac{8}{3}} d x - \int \frac{2}{3 x} d x

\int 27 x^{\frac{8}{3}} d x = \frac{81}{11} x^{\frac{11}{3}}

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

= \frac{81}{11} x^{\frac{11}{3}} - 3 x^{\frac{2}{3}}

解答补常数

= \frac{81}{11} x^{\frac{11}{3}} - 3 x^{\frac{2}{3}} + C

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