integral of 2piy(8-y^{3/2})

解答

\int 2 π y (8 - y^{\frac{3}{2}}) d y

2 π (4 y^{2} - \frac{2}{7} y^{\frac{7}{2}}) + C

求解步骤

\int 2 π y (8 - y^{\frac{3}{2}}) d y

提出常数:

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

= 2 π \cdot \int y (8 - y^{\frac{3}{2}}) d y

乘开

y (8 - y^{\frac{3}{2}}) : 8 y - y^{\frac{5}{2}}

= 2 π \cdot \int 8 y - y^{\frac{5}{2}} d y

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

= 2 π (\int 8 y d y - \int y^{\frac{5}{2}} d y)

\int 8 y d y = 4 y^{2}

\int y^{\frac{5}{2}} d y = \frac{2}{7} y^{\frac{7}{2}}

= 2 π (4 y^{2} - \frac{2}{7} y^{\frac{7}{2}})

解答补常数

= 2 π (4 y^{2} - \frac{2}{7} y^{\frac{7}{2}}) + C

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t an g e n t f (x) = x^{3} + 2 x, at x = 1

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