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受欢迎的微积分 >

integral of x(a-x)^{5/2}

解答

\int x (a - x)^{\frac{5}{2}} d x

解答

- \frac{2}{63} (7 x + 2 a) (- x + a)^{\frac{7}{2}} + C

求解步骤

\int x (a - x)^{\frac{5}{2}} d x

使用换元积分法

= \int - \frac{2 u ^{\frac{2}{5}} ( - u ^{\frac{2}{5}} + a )}{5} d u

提出常数:

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

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

乘开

u^{\frac{2}{5}} (- u^{\frac{2}{5}} + a) : - u^{\frac{4}{5}} + a u^{\frac{2}{5}}

= - \frac{2}{5} \cdot \int - u^{\frac{4}{5}} + a u^{\frac{2}{5}} 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{2}{5} (- \int u^{\frac{4}{5}} d u + \int a u^{\frac{2}{5}} d u)

\int u^{\frac{4}{5}} d u = \frac{5}{9} u^{\frac{9}{5}}

\int a u^{\frac{2}{5}} d u = a \frac{5}{7} u^{\frac{7}{5}}

= - \frac{2}{5} (- \frac{5}{9} u^{\frac{9}{5}} + a \frac{5}{7} u^{\frac{7}{5}})

u = (a - x)^{\frac{5}{2}}

代回

= - \frac{2}{5} (- \frac{5}{9} ((a - x)^{\frac{5}{2}})^{\frac{9}{5}} + a \frac{5}{7} ((a - x)^{\frac{5}{2}})^{\frac{7}{5}})

化简

- \frac{2}{5} (- \frac{5}{9} ((a - x)^{\frac{5}{2}})^{\frac{9}{5}} + a \frac{5}{7} ((a - x)^{\frac{5}{2}})^{\frac{7}{5}}) : - \frac{2}{63} (7 x + 2 a) (- x + a)^{\frac{7}{2}}

= - \frac{2}{63} (7 x + 2 a) (- x + a)^{\frac{7}{2}}

解答补常数

= - \frac{2}{63} (7 x + 2 a) (- x + a)^{\frac{7}{2}} + C

作图

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