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受欢迎的微积分 >

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

解答

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

解答

\frac{3}{88} (8 x - 9) (x + 3)^{\frac{8}{3}} + C

求解步骤

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

使用换元积分法

= \int \frac{5 u - 3}{5 u ^{\frac{7}{15}}} d u

提出常数:

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

= \frac{1}{5} \cdot \int \frac{5 u - 3}{u ^{\frac{7}{15}}} d u

乘开

\frac{5 u - 3}{u ^{\frac{7}{15}}} : \frac{1}{u ^{\frac{4}{15}}} - \frac{3}{u ^{\frac{7}{15}}}

= \frac{1}{5} \cdot \int \frac{1}{u ^{\frac{4}{15}}} - \frac{3}{u ^{\frac{7}{15}}} 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}{5} (\int \frac{1}{u ^{\frac{4}{15}}} d u - \int \frac{3}{u ^{\frac{7}{15}}} d u)

\int \frac{1}{u ^{\frac{4}{15}}} d u = \frac{15}{11} u^{\frac{11}{15}}

\int \frac{3}{u ^{\frac{7}{15}}} d u = \frac{45}{8} u^{\frac{8}{15}}

= \frac{1}{5} (\frac{15}{11} u^{\frac{11}{15}} - \frac{45}{8} u^{\frac{8}{15}})

u = (x + 3)^{5}

代回

= \frac{1}{5} (\frac{15}{11} ((x + 3)^{5})^{\frac{11}{15}} - \frac{45}{8} ((x + 3)^{5})^{\frac{8}{15}})

化简

\frac{1}{5} (\frac{15}{11} ((x + 3)^{5})^{\frac{11}{15}} - \frac{45}{8} ((x + 3)^{5})^{\frac{8}{15}}) : \frac{3}{88} (8 x - 9) (x + 3)^{\frac{8}{3}}

= \frac{3}{88} (8 x - 9) (x + 3)^{\frac{8}{3}}

解答补常数

= \frac{3}{88} (8 x - 9) (x + 3)^{\frac{8}{3}} + C

作图

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