integral of (18x^2+3)(6x^3+3x)^6

解

\int (18 x^{2} + 3) (6 x^{3} + 3 x)^{6} d x

\frac{279936 x ^{21}}{7} + 139968 x^{19} + 209952 x^{17} + 174960 x^{15} + 87480 x^{13} + 26244 x^{11} + 4374 x^{9} + \frac{2187 x ^{7}}{7} + C

解答ステップ

\int (18 x^{2} + 3) (6 x^{3} + 3 x)^{6} d x

拡張

(18 x^{2} + 3) (6 x^{3} + 3 x)^{6} : 839808 x^{20} + 2659392 x^{18} + 3569184 x^{16} + 2624400 x^{14} + 1137240 x^{12} + 288684 x^{10} + 39366 x^{8} + 2187 x^{6}

= \int 839808 x^{20} + 2659392 x^{18} + 3569184 x^{16} + 2624400 x^{14} + 1137240 x^{12} + 288684 x^{10} + 39366 x^{8} + 2187 x^{6} 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 839808 x^{20} d x + \int 2659392 x^{18} d x + \int 3569184 x^{16} d x + \int 2624400 x^{14} d x + \int 1137240 x^{12} d x + \int 288684 x^{10} d x + \int 39366 x^{8} d x + \int 2187 x^{6} d x

\int 839808 x^{20} d x = \frac{279936 x ^{21}}{7}

\int 2659392 x^{18} d x = 139968 x^{19}

\int 3569184 x^{16} d x = 209952 x^{17}

\int 2624400 x^{14} d x = 174960 x^{15}

\int 1137240 x^{12} d x = 87480 x^{13}

\int 288684 x^{10} d x = 26244 x^{11}

\int 39366 x^{8} d x = 4374 x^{9}

\int 2187 x^{6} d x = \frac{2187 x ^{7}}{7}

= \frac{279936 x ^{21}}{7} + 139968 x^{19} + 209952 x^{17} + 174960 x^{15} + 87480 x^{13} + 26244 x^{11} + 4374 x^{9} + \frac{2187 x ^{7}}{7}

定数を解答に追加する

= \frac{279936 x ^{21}}{7} + 139968 x^{19} + 209952 x^{17} + 174960 x^{15} + 87480 x^{13} + 26244 x^{11} + 4374 x^{9} + \frac{2187 x ^{7}}{7} + C