integral from 0 to 4 of 1/((2-3x)^{1/3)}

解答

\int_{0}^{4} \frac{1}{( 2 - 3 x ) ^{\frac{1}{3}}} d x

\frac{1}{2 ^{\frac{1}{3}}} - (\frac{25}{2})^{\frac{1}{3}}

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求解步骤

\int_{0}^{4} \frac{1}{( 2 - 3 x ) ^{\frac{1}{3}}} d x

找到无定义的点（奇点）:

x = \frac{2}{3}

如果存在

b, a < b < c, f (b) =

无定义

, \int_{a}^{c} f (x) d x = \int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x

= \int_{0}^{\frac{2}{3}} \frac{1}{( 2 - 3 x ) ^{\frac{1}{3}}} d x + \int_{\frac{2}{3}}^{4} \frac{1}{( 2 - 3 x ) ^{\frac{1}{3}}} d x

\int_{0}^{\frac{2}{3}} \frac{1}{( 2 - 3 x ) ^{\frac{1}{3}}} d x = \frac{1}{2 ^{\frac{1}{3}}}

\int_{\frac{2}{3}}^{4} \frac{1}{( 2 - 3 x ) ^{\frac{1}{3}}} d x = - \frac{( - 1 ) ^{\frac{2}{3}} \cdot 5 ^{\frac{2}{3}}}{2 ^{\frac{1}{3}}}

= \frac{1}{2 ^{\frac{1}{3}}} - \frac{( - 1 ) ^{\frac{2}{3}} \cdot 5 ^{\frac{2}{3}}}{2 ^{\frac{1}{3}}}

化简

\frac{1}{2 ^{\frac{1}{3}}} - \frac{( - 1 ) ^{\frac{2}{3}} \cdot 5 ^{\frac{2}{3}}}{2 ^{\frac{1}{3}}} : \frac{1}{2 ^{\frac{1}{3}}} - (\frac{25}{2})^{\frac{1}{3}}

= \frac{1}{2 ^{\frac{1}{3}}} - (\frac{25}{2})^{\frac{1}{3}}

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