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受欢迎的微积分 >

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

解答

\int 3 (x^{3} + 1) x^{5} d x

解答

\frac{1}{28} (4 (x^{3} + 1)^{\frac{7}{3}} - 7 (x^{3} + 1)^{\frac{4}{3}}) + C

求解步骤

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

使用换元积分法

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

乘开

(u - 1) 3 u : u^{\frac{4}{3}} - 3 u

= \frac{1}{3} \cdot \int u^{\frac{4}{3}} - 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{4}{3}} d u - \int 3 u d u)

\int u^{\frac{4}{3}} d u = \frac{3}{7} u^{\frac{7}{3}}

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

= \frac{1}{3} (\frac{3}{7} u^{\frac{7}{3}} - \frac{3}{4} u^{\frac{4}{3}})

u = x^{3} + 1

代回

= \frac{1}{3} (\frac{3}{7} (x^{3} + 1)^{\frac{7}{3}} - \frac{3}{4} (x^{3} + 1)^{\frac{4}{3}})

化简

\frac{1}{3} (\frac{3}{7} (x^{3} + 1)^{\frac{7}{3}} - \frac{3}{4} (x^{3} + 1)^{\frac{4}{3}}) : \frac{1}{28} (4 (x^{3} + 1)^{\frac{7}{3}} - 7 (x^{3} + 1)^{\frac{4}{3}})

= \frac{1}{28} (4 (x^{3} + 1)^{\frac{7}{3}} - 7 (x^{3} + 1)^{\frac{4}{3}})

解答补常数

= \frac{1}{28} (4 (x^{3} + 1)^{\frac{7}{3}} - 7 (x^{3} + 1)^{\frac{4}{3}}) + C

作图

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