integral of 9x^3(2x^4-6)^7

解

\int 9 x^{3} (2 x^{4} - 6)^{7} d x

9 (4 x^{32} - 96 x^{28} + 1008 x^{24} - 6048 x^{20} + 22680 x^{16} - 54432 x^{12} + 81648 x^{8} - 69984 x^{4}) + C

解答ステップ

\int 9 x^{3} (2 x^{4} - 6)^{7} d x

定数を除く:

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

= 9 \cdot \int x^{3} (2 x^{4} - 6)^{7} d x

拡張

x^{3} (2 x^{4} - 6)^{7} : 128 x^{31} - 2688 x^{27} + 24192 x^{23} - 120960 x^{19} + 362880 x^{15} - 653184 x^{11} + 653184 x^{7} - 279936 x^{3}

= 9 \cdot \int 128 x^{31} - 2688 x^{27} + 24192 x^{23} - 120960 x^{19} + 362880 x^{15} - 653184 x^{11} + 653184 x^{7} - 279936 x^{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

= 9 (\int 128 x^{31} d x - \int 2688 x^{27} d x + \int 24192 x^{23} d x - \int 120960 x^{19} d x + \int 362880 x^{15} d x - \int 653184 x^{11} d x + \int 653184 x^{7} d x - \int 279936 x^{3} d x)

\int 128 x^{31} d x = 4 x^{32}

\int 2688 x^{27} d x = 96 x^{28}

\int 24192 x^{23} d x = 1008 x^{24}

\int 120960 x^{19} d x = 6048 x^{20}

\int 362880 x^{15} d x = 22680 x^{16}

\int 653184 x^{11} d x = 54432 x^{12}

\int 653184 x^{7} d x = 81648 x^{8}

\int 279936 x^{3} d x = 69984 x^{4}

= 9 (4 x^{32} - 96 x^{28} + 1008 x^{24} - 6048 x^{20} + 22680 x^{16} - 54432 x^{12} + 81648 x^{8} - 69984 x^{4})

定数を解答に追加する

= 9 (4 x^{32} - 96 x^{28} + 1008 x^{24} - 6048 x^{20} + 22680 x^{16} - 54432 x^{12} + 81648 x^{8} - 69984 x^{4}) + C