integral from x^4 to x^5 of (-4x-2)^3

解

\int_{x^{4}}^{x^{5}} (- 4 x - 2)^{3} d x

- 16 x^{20} + 16 x^{16} - 32 x^{15} + 32 x^{12} - 24 x^{10} + 24 x^{8} - 8 x^{5} + 8 x^{4}

解答ステップ

\int_{x^{4}}^{x^{5}} (- 4 x - 2)^{3} d x

拡張

(- 4 x - 2)^{3} : - 64 x^{3} - 96 x^{2} - 48 x - 8

= \int_{x^{4}}^{x^{5}} - 64 x^{3} - 96 x^{2} - 48 x - 8 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_{x^{4}}^{x^{5}} 64 x^{3} d x - \int_{x^{4}}^{x^{5}} 96 x^{2} d x - \int_{x^{4}}^{x^{5}} 48 x d x - \int_{x^{4}}^{x^{5}} 8 d x

\int_{x^{4}}^{x^{5}} 64 x^{3} d x = 16 (x^{20} - x^{16})

\int_{x^{4}}^{x^{5}} 96 x^{2} d x = 32 (x^{15} - x^{12})

\int_{x^{4}}^{x^{5}} 48 x d x = 24 (x^{10} - x^{8})

\int_{x^{4}}^{x^{5}} 8 d x = 8 x^{5} - 8 x^{4}

= - 16 (x^{20} - x^{16}) - 32 (x^{15} - x^{12}) - 24 (x^{10} - x^{8}) - (8 x^{5} - 8 x^{4})

簡素化

- 16 (x^{20} - x^{16}) - 32 (x^{15} - x^{12}) - 24 (x^{10} - x^{8}) - (8 x^{5} - 8 x^{4}) : - 16 x^{20} + 16 x^{16} - 32 x^{15} + 32 x^{12} - 24 x^{10} + 24 x^{8} - 8 x^{5} + 8 x^{4}

= - 16 x^{20} + 16 x^{16} - 32 x^{15} + 32 x^{12} - 24 x^{10} + 24 x^{8} - 8 x^{5} + 8 x^{4}