integral from 0 to 9 of 1/((x-1)^{1/3)}

解

\int_{0}^{9} \frac{1}{( x - 1 ) ^{\frac{1}{3}}} d x

\frac{9}{2}

十進法表記

4.5

解答ステップ

\int_{0}^{9} \frac{1}{( x - 1 ) ^{\frac{1}{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{1}{3}}} d x + \int_{1}^{9} \frac{1}{( x - 1 ) ^{\frac{1}{3}}} d x

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

\int_{1}^{9} \frac{1}{( x - 1 ) ^{\frac{1}{3}}} d x = 6

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

簡素化

- \frac{3 ( - 1 ) ^{\frac{2}{3}}}{2} + 6 : \frac{9}{2}

= \frac{9}{2}