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

解答

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

3 \cdot 4^{\frac{1}{3}} + 3 \cdot 2^{\frac{1}{3}}

十进制

8.54196 \dots

求解步骤

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

找到无定义的点（奇点）:

x = 2

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

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

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

= - 3 (- 1)^{\frac{1}{3}} \cdot 4^{\frac{1}{3}} + 3 \cdot 2^{\frac{1}{3}}

化简

- 3 (- 1)^{\frac{1}{3}} \cdot 4^{\frac{1}{3}} + 3 \cdot 2^{\frac{1}{3}} : 3 \cdot 4^{\frac{1}{3}} + 3 \cdot 2^{\frac{1}{3}}

= 3 \cdot 4^{\frac{1}{3}} + 3 \cdot 2^{\frac{1}{3}}

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