integral from 1 to 11 of 1/((x-3)^{2/3)}

解答

\int_{1}^{11} \frac{1}{( x - 3 ) ^{\frac{2}{3}}} d x

3 \cdot 2^{\frac{1}{3}} + 6

十进制

9.77976 \dots

求解步骤

\int_{1}^{11} \frac{1}{( x - 3 ) ^{\frac{2}{3}}} d x

找到无定义的点（奇点）:

x = 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_{1}^{3} \frac{1}{( x - 3 ) ^{\frac{2}{3}}} d x + \int_{3}^{11} \frac{1}{( x - 3 ) ^{\frac{2}{3}}} d x

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

\int_{3}^{11} \frac{1}{( x - 3 ) ^{\frac{2}{3}}} d x = 6

= - 3 (- 1)^{\frac{1}{3}} \cdot 2^{\frac{1}{3}} + 6

化简

- 3 (- 1)^{\frac{1}{3}} \cdot 2^{\frac{1}{3}} + 6 : 3 \cdot 2^{\frac{1}{3}} + 6

= 3 \cdot 2^{\frac{1}{3}} + 6

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