integral from-8 to 1 of x^{-2/3}

解答

\int_{- 8}^{1} x^{- \frac{2}{3}} d x

9

求解步骤

\int_{- 8}^{1} x^{- \frac{2}{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_{- 8}^{0} x^{- \frac{2}{3}} d x + \int_{0}^{1} x^{- \frac{2}{3}} d x

\int_{- 8}^{0} x^{- \frac{2}{3}} d x = - 6 (- 1)^{\frac{1}{3}}

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

= - 6 (- 1)^{\frac{1}{3}} + 3

化简

- 6 (- 1)^{\frac{1}{3}} + 3 : 9

= 9

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