integral of (4x^2-9x)^5(8x-9)

解

\int (4 x^{2} - 9 x)^{5} (8 x - 9) d x

\frac{2048 x ^{12}}{3} - 9216 x^{11} + 51840 x^{10} - 155520 x^{9} + 262440 x^{8} - 236196 x^{7} + \frac{177147 x ^{6}}{2} + C

解答ステップ

\int (4 x^{2} - 9 x)^{5} (8 x - 9) d x

拡張

(4 x^{2} - 9 x)^{5} (8 x - 9) : 8192 x^{11} - 101376 x^{10} + 518400 x^{9} - 1399680 x^{8} + 2099520 x^{7} - 1653372 x^{6} + 531441 x^{5}

= \int 8192 x^{11} - 101376 x^{10} + 518400 x^{9} - 1399680 x^{8} + 2099520 x^{7} - 1653372 x^{6} + 531441 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

= \int 8192 x^{11} d x - \int 101376 x^{10} d x + \int 518400 x^{9} d x - \int 1399680 x^{8} d x + \int 2099520 x^{7} d x - \int 1653372 x^{6} d x + \int 531441 x^{5} d x

\int 8192 x^{11} d x = \frac{2048 x ^{12}}{3}

\int 101376 x^{10} d x = 9216 x^{11}

\int 518400 x^{9} d x = 51840 x^{10}

\int 1399680 x^{8} d x = 155520 x^{9}

\int 2099520 x^{7} d x = 262440 x^{8}

\int 1653372 x^{6} d x = 236196 x^{7}

\int 531441 x^{5} d x = \frac{177147 x ^{6}}{2}

= \frac{2048 x ^{12}}{3} - 9216 x^{11} + 51840 x^{10} - 155520 x^{9} + 262440 x^{8} - 236196 x^{7} + \frac{177147 x ^{6}}{2}

定数を解答に追加する

= \frac{2048 x ^{12}}{3} - 9216 x^{11} + 51840 x^{10} - 155520 x^{9} + 262440 x^{8} - 236196 x^{7} + \frac{177147 x ^{6}}{2} + C