integral from 5 to-5 of 1/(|x|^{2/3)}

解

\int_{5}^{- 5} \frac{1}{∣ x ∣ ^{\frac{2}{3}}} d x

- 6 \cdot 5^{\frac{1}{3}}

十進法表記

- 10.25985 \dots

解答ステップ

\int_{5}^{- 5} \frac{1}{∣ x ∣ ^{\frac{2}{3}}} d x

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{- 5}^{5} \frac{1}{∣ x ∣ ^{\frac{2}{3}}} d x

絶対数を排除する

= - (\int_{- 5}^{0} \frac{1}{( - x ) ^{\frac{2}{3}}} d x + \int_{0}^{5} \frac{1}{x ^{\frac{2}{3}}} d x)

\int_{- 5}^{0} \frac{1}{( - x ) ^{\frac{2}{3}}} d x = - 3 (- 1)^{\frac{1}{3}} \cdot 5^{\frac{1}{3}}

\int_{0}^{5} \frac{1}{x ^{\frac{2}{3}}} d x = 3 \cdot 5^{\frac{1}{3}}

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

簡素化

- (- 3 (- 1)^{\frac{1}{3}} \cdot 5^{\frac{1}{3}} + 3 \cdot 5^{\frac{1}{3}}) : - 6 \cdot 5^{\frac{1}{3}}

= - 6 \cdot 5^{\frac{1}{3}}