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受欢迎的微积分 >

integral of (x^5)/(\sqrt[3]{1+x^3)}

解答

\int \frac{x ^{5}}{3 1 + x ^{3}} d x

解答

\frac{1}{10} (2 x^{3} - 3) (x^{3} + 1)^{\frac{2}{3}} + C

求解步骤

\int \frac{x ^{5}}{3 1 + x ^{3}} d x

使用换元积分法

= \int \frac{u - 1}{3 3 u} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int \frac{u - 1}{3 u} d u

乘开

\frac{u - 1}{3 u} : u^{\frac{2}{3}} - \frac{1}{3 u}

= \frac{1}{3} \cdot \int u^{\frac{2}{3}} - \frac{1}{3 u} d u

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

= \frac{1}{3} (\int u^{\frac{2}{3}} d u - \int \frac{1}{3 u} d u)

\int u^{\frac{2}{3}} d u = \frac{3}{5} u^{\frac{5}{3}}

\int \frac{1}{3 u} d u = \frac{3}{2} u^{\frac{2}{3}}

= \frac{1}{3} (\frac{3}{5} u^{\frac{5}{3}} - \frac{3}{2} u^{\frac{2}{3}})

u = 1 + x^{3}

代回

= \frac{1}{3} (\frac{3}{5} (1 + x^{3})^{\frac{5}{3}} - \frac{3}{2} (1 + x^{3})^{\frac{2}{3}})

化简

\frac{1}{3} (\frac{3}{5} (1 + x^{3})^{\frac{5}{3}} - \frac{3}{2} (1 + x^{3})^{\frac{2}{3}}) : \frac{1}{10} (2 x^{3} - 3) (x^{3} + 1)^{\frac{2}{3}}

= \frac{1}{10} (2 x^{3} - 3) (x^{3} + 1)^{\frac{2}{3}}

解答补常数

= \frac{1}{10} (2 x^{3} - 3) (x^{3} + 1)^{\frac{2}{3}} + C

作图

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