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

解

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

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

十進法表記

3.66337 \dots

解答ステップ

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

未定義の (特異) 点を求める:

x = \frac{1}{2}

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

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

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

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

簡素化

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

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