integral from-1 to 0 of 384x^5(8x^6-3)^3

解

\int_{- 1}^{0} 384 x^{5} (8 x^{6} - 3)^{3} d x

- 1088

解答ステップ

\int_{- 1}^{0} 384 x^{5} (8 x^{6} - 3)^{3} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 384 \cdot \int_{- 1}^{0} x^{5} (8 x^{6} - 3)^{3} d x

拡張

x^{5} (8 x^{6} - 3)^{3} : 512 x^{23} - 576 x^{17} + 216 x^{11} - 27 x^{5}

= 384 \cdot \int_{- 1}^{0} 512 x^{23} - 576 x^{17} + 216 x^{11} - 27 x^{5} 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

= 384 (\int_{- 1}^{0} 512 x^{23} d x - \int_{- 1}^{0} 576 x^{17} d x + \int_{- 1}^{0} 216 x^{11} d x - \int_{- 1}^{0} 27 x^{5} d x)

\int_{- 1}^{0} 512 x^{23} d x = - \frac{64}{3}

\int_{- 1}^{0} 576 x^{17} d x = - 32

\int_{- 1}^{0} 216 x^{11} d x = - 18

\int_{- 1}^{0} 27 x^{5} d x = - \frac{9}{2}

= 384 (- \frac{64}{3} - (- 32) - 18 - (- \frac{9}{2}))

簡素化

384 (- \frac{64}{3} - (- 32) - 18 - (- \frac{9}{2})) : - 1088

= - 1088