integral from 1 to 5 of 1/((x-2)^{1/5)}

解

\int_{1}^{5} \frac{1}{( x - 2 ) ^{\frac{1}{5}}} d x

- \frac{5}{4} + \frac{5 \cdot 3 ^{\frac{4}{5}}}{4}

十進法表記

1.76028 \dots

解答ステップ

\int_{1}^{5} \frac{1}{( x - 2 ) ^{\frac{1}{5}}} d x

未定義の (特異) 点を求める:

x = 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_{1}^{2} \frac{1}{( x - 2 ) ^{\frac{1}{5}}} d x + \int_{2}^{5} \frac{1}{( x - 2 ) ^{\frac{1}{5}}} d x

\int_{1}^{2} \frac{1}{( x - 2 ) ^{\frac{1}{5}}} d x = - \frac{5 ( - 1 ) ^{\frac{4}{5}}}{4}

\int_{2}^{5} \frac{1}{( x - 2 ) ^{\frac{1}{5}}} d x = \frac{5 \cdot 3 ^{\frac{4}{5}}}{4}

= - \frac{5 ( - 1 ) ^{\frac{4}{5}}}{4} + \frac{5 \cdot 3 ^{\frac{4}{5}}}{4}

5 (- 1)^{\frac{4}{5}} = 5

= - \frac{5}{4} + \frac{5 \cdot 3 ^{\frac{4}{5}}}{4}