integral from-1 to 1000 of x^{-1/3}

解答

\int_{- 1}^{1000} x^{- \frac{1}{3}} d x

\frac{297}{2}

十进制

148.5

求解步骤

\int_{- 1}^{1000} x^{- \frac{1}{3}} d x

找到无定义的点（奇点）:

x = 0

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_{- 1}^{0} x^{- \frac{1}{3}} d x + \int_{0}^{1000} x^{- \frac{1}{3}} d x

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

\int_{0}^{1000} x^{- \frac{1}{3}} d x = 150

= - \frac{3 ( - 1 ) ^{\frac{2}{3}}}{2} + 150

化简

- \frac{3 ( - 1 ) ^{\frac{2}{3}}}{2} + 150 : \frac{297}{2}

= \frac{297}{2}

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