integral of (1-3x+x^3)\sqrt[3]{x}

解答

\int (1 - 3 x + x^{3}) 3 x d x

\frac{3}{4} x^{\frac{4}{3}} - \frac{9}{7} x^{\frac{7}{3}} + \frac{3}{13} x^{\frac{13}{3}} + C

求解步骤

\int (1 - 3 x + x^{3}) 3 x d x

乘开

(1 - 3 x + x^{3}) 3 x : 3 x - 3 x^{\frac{4}{3}} + x^{\frac{10}{3}}

= \int 3 x - 3 x^{\frac{4}{3}} + x^{\frac{10}{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 3 x d x - \int 3 x^{\frac{4}{3}} d x + \int x^{\frac{10}{3}} d x

\int 3 x d x = \frac{3}{4} x^{\frac{4}{3}}

\int 3 x^{\frac{4}{3}} d x = \frac{9}{7} x^{\frac{7}{3}}

\int x^{\frac{10}{3}} d x = \frac{3}{13} x^{\frac{13}{3}}

= \frac{3}{4} x^{\frac{4}{3}} - \frac{9}{7} x^{\frac{7}{3}} + \frac{3}{13} x^{\frac{13}{3}}

解答补常数

= \frac{3}{4} x^{\frac{4}{3}} - \frac{9}{7} x^{\frac{7}{3}} + \frac{3}{13} x^{\frac{13}{3}} + C

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