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

解

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

\frac{276 - 705 \cdot 2 ^{\frac{2}{3}} \cdot 5 ^{\frac{1}{3}}}{28 \cdot 2 ^{\frac{2}{3}}}

十進法表記

- 36.84514 \dots

解答ステップ

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

拡張

x^{\frac{1}{3}} (- 1 - 2 x) : - x^{\frac{1}{3}} - 2 x^{\frac{4}{3}}

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

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int_{2}^{5} x^{\frac{1}{3}} d x = \frac{15 \cdot 5 ^{\frac{1}{3}}}{4} - \frac{3}{2 ^{\frac{2}{3}}}

\int_{2}^{5} 2 x^{\frac{4}{3}} d x = \frac{2 ( 75 \cdot 5 ^{\frac{1}{3}} - 12 \cdot 2 ^{\frac{1}{3}} )}{7}

= - (\frac{15 \cdot 5 ^{\frac{1}{3}}}{4} - \frac{3}{2 ^{\frac{2}{3}}}) - \frac{2 ( 75 \cdot 5 ^{\frac{1}{3}} - 12 \cdot 2 ^{\frac{1}{3}} )}{7}

簡素化

- (\frac{15 \cdot 5 ^{\frac{1}{3}}}{4} - \frac{3}{2 ^{\frac{2}{3}}}) - \frac{2 ( 75 \cdot 5 ^{\frac{1}{3}} - 12 \cdot 2 ^{\frac{1}{3}} )}{7} : \frac{276 - 705 \cdot 2 ^{\frac{2}{3}} \cdot 5 ^{\frac{1}{3}}}{28 \cdot 2 ^{\frac{2}{3}}}

= \frac{276 - 705 \cdot 2 ^{\frac{2}{3}} \cdot 5 ^{\frac{1}{3}}}{28 \cdot 2 ^{\frac{2}{3}}}