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

解

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

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

十進法表記

6.77976 \dots

解答ステップ

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

未定義の (特異) 点を求める:

x = 1

存在する場合は

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

未定義

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

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

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

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

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

簡素化

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

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