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受欢迎的微积分 >

integral of x(5-x)^9

解答

\int x (5 - x)^{9} d x

解答

- \frac{1}{22} (- x + 5)^{10} (2 x + 1) + C

求解步骤

\int x (5 - x)^{9} d x

使用换元积分法

= \int - \frac{u ^{\frac{1}{9}} ( - 9 u + 5 )}{9} d u

提出常数:

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

= - \frac{1}{9} \cdot \int u^{\frac{1}{9}} (- 9 u + 5) d u

乘开

u^{\frac{1}{9}} (- 9 u + 5) : - u^{\frac{2}{9}} + 5 u^{\frac{1}{9}}

= - \frac{1}{9} \cdot \int - u^{\frac{2}{9}} + 5 u^{\frac{1}{9}} 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}{9} (- \int u^{\frac{2}{9}} d u + \int 5 u^{\frac{1}{9}} d u)

\int u^{\frac{2}{9}} d u = \frac{9}{11} u^{\frac{11}{9}}

\int 5 u^{\frac{1}{9}} d u = \frac{9}{2} u^{\frac{10}{9}}

= - \frac{1}{9} (- \frac{9}{11} u^{\frac{11}{9}} + \frac{9}{2} u^{\frac{10}{9}})

u = (5 - x)^{9}

代回

= - \frac{1}{9} (- \frac{9}{11} ((5 - x)^{9})^{\frac{11}{9}} + \frac{9}{2} ((5 - x)^{9})^{\frac{10}{9}})

化简

- \frac{1}{9} (- \frac{9}{11} ((5 - x)^{9})^{\frac{11}{9}} + \frac{9}{2} ((5 - x)^{9})^{\frac{10}{9}}) : - \frac{1}{22} (- x + 5)^{10} (2 x + 1)

= - \frac{1}{22} (- x + 5)^{10} (2 x + 1)

解答补常数

= - \frac{1}{22} (- x + 5)^{10} (2 x + 1) + C

作图

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