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受欢迎的微积分 >

integral of x(x+8)^7

解答

\int x (x + 8)^{7} d x

解答

\frac{1}{9} (x + 8)^{8} (x - 1) + C

求解步骤

\int x (x + 8)^{7} d x

使用换元积分法

= \int \frac{7 u ( 7 u - 8 )}{7} d u

提出常数:

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

= \frac{1}{7} \cdot \int 7 u (7 u - 8) d u

乘开

7 u (7 u - 8) : u^{\frac{2}{7}} - 8 7 u

= \frac{1}{7} \cdot \int u^{\frac{2}{7}} - 8 7 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}{7} (\int u^{\frac{2}{7}} d u - \int 8 7 u d u)

\int u^{\frac{2}{7}} d u = \frac{7}{9} u^{\frac{9}{7}}

\int 8 7 u d u = 7 u^{\frac{8}{7}}

= \frac{1}{7} (\frac{7}{9} u^{\frac{9}{7}} - 7 u^{\frac{8}{7}})

u = (x + 8)^{7}

代回

= \frac{1}{7} (\frac{7}{9} ((x + 8)^{7})^{\frac{9}{7}} - 7 ((x + 8)^{7})^{\frac{8}{7}})

化简

\frac{1}{7} (\frac{7}{9} ((x + 8)^{7})^{\frac{9}{7}} - 7 ((x + 8)^{7})^{\frac{8}{7}}) : \frac{1}{9} (x + 8)^{8} (x - 1)

= \frac{1}{9} (x + 8)^{8} (x - 1)

解答补常数

= \frac{1}{9} (x + 8)^{8} (x - 1) + C

作图

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