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受欢迎的微积分 >

integral of (x+2)sqrt(8-x)

解答

\int (x + 2) 8 - x d x

解答

\frac{2}{5} (8 - x)^{\frac{5}{2}} - \frac{20}{3} (8 - x)^{\frac{3}{2}} + C

求解步骤

\int (x + 2) 8 - x d x

使用换元积分法

= \int - u (- u + 10) d u

提出常数:

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

= - \int u (- u + 10) d u

乘开

u (- u + 10) : - u^{\frac{3}{2}} + 10 u

= - \int - u^{\frac{3}{2}} + 10 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

= - (- \int u^{\frac{3}{2}} d u + \int 10 u d u)

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

\int 10 u d u = \frac{20}{3} u^{\frac{3}{2}}

= - (- \frac{2}{5} u^{\frac{5}{2}} + \frac{20}{3} u^{\frac{3}{2}})

u = 8 - x

代回

= - (- \frac{2}{5} (8 - x)^{\frac{5}{2}} + \frac{20}{3} (8 - x)^{\frac{3}{2}})

化简

= \frac{2}{5} (8 - x)^{\frac{5}{2}} - \frac{20}{3} (8 - x)^{\frac{3}{2}}

解答补常数

= \frac{2}{5} (8 - x)^{\frac{5}{2}} - \frac{20}{3} (8 - x)^{\frac{3}{2}} + C

作图

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