integral of (x^2-6x)^5(x-3)

解

\int (x^{2} - 6 x)^{5} (x - 3) d x

\frac{x ^{12}}{12} - 3 x^{11} + 45 x^{10} - 360 x^{9} + 1620 x^{8} - 3888 x^{7} + 3888 x^{6} + C

解答ステップ

\int (x^{2} - 6 x)^{5} (x - 3) d x

拡張

(x^{2} - 6 x)^{5} (x - 3) : x^{11} - 33 x^{10} + 450 x^{9} - 3240 x^{8} + 12960 x^{7} - 27216 x^{6} + 23328 x^{5}

= \int x^{11} - 33 x^{10} + 450 x^{9} - 3240 x^{8} + 12960 x^{7} - 27216 x^{6} + 23328 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 x^{11} d x - \int 33 x^{10} d x + \int 450 x^{9} d x - \int 3240 x^{8} d x + \int 12960 x^{7} d x - \int 27216 x^{6} d x + \int 23328 x^{5} d x

\int x^{11} d x = \frac{x ^{12}}{12}

\int 33 x^{10} d x = 3 x^{11}

\int 450 x^{9} d x = 45 x^{10}

\int 3240 x^{8} d x = 360 x^{9}

\int 12960 x^{7} d x = 1620 x^{8}

\int 27216 x^{6} d x = 3888 x^{7}

\int 23328 x^{5} d x = 3888 x^{6}

= \frac{x ^{12}}{12} - 3 x^{11} + 45 x^{10} - 360 x^{9} + 1620 x^{8} - 3888 x^{7} + 3888 x^{6}

定数を解答に追加する

= \frac{x ^{12}}{12} - 3 x^{11} + 45 x^{10} - 360 x^{9} + 1620 x^{8} - 3888 x^{7} + 3888 x^{6} + C