integral of x(2x-1)^9

解

\int x (2 x - 1)^{9} d x

\frac{512 x ^{11}}{11} - \frac{1152 x ^{10}}{5} + 512 x^{9} - 672 x^{8} + 576 x^{7} - 336 x^{6} + \frac{672 x ^{5}}{5} - 36 x^{4} + 6 x^{3} - \frac{x ^{2}}{2} + C

解答ステップ

\int x (2 x - 1)^{9} d x

拡張

x (2 x - 1)^{9} : 512 x^{10} - 2304 x^{9} + 4608 x^{8} - 5376 x^{7} + 4032 x^{6} - 2016 x^{5} + 672 x^{4} - 144 x^{3} + 18 x^{2} - x

= \int 512 x^{10} - 2304 x^{9} + 4608 x^{8} - 5376 x^{7} + 4032 x^{6} - 2016 x^{5} + 672 x^{4} - 144 x^{3} + 18 x^{2} - x 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 512 x^{10} d x - \int 2304 x^{9} d x + \int 4608 x^{8} d x - \int 5376 x^{7} d x + \int 4032 x^{6} d x - \int 2016 x^{5} d x + \int 672 x^{4} d x - \int 144 x^{3} d x + \int 18 x^{2} d x - \int x d x

\int 512 x^{10} d x = \frac{512 x ^{11}}{11}

\int 2304 x^{9} d x = \frac{1152 x ^{10}}{5}

\int 4608 x^{8} d x = 512 x^{9}

\int 5376 x^{7} d x = 672 x^{8}

\int 4032 x^{6} d x = 576 x^{7}

\int 2016 x^{5} d x = 336 x^{6}

\int 672 x^{4} d x = \frac{672 x ^{5}}{5}

\int 144 x^{3} d x = 36 x^{4}

\int 18 x^{2} d x = 6 x^{3}

\int x d x = \frac{x ^{2}}{2}

= \frac{512 x ^{11}}{11} - \frac{1152 x ^{10}}{5} + 512 x^{9} - 672 x^{8} + 576 x^{7} - 336 x^{6} + \frac{672 x ^{5}}{5} - 36 x^{4} + 6 x^{3} - \frac{x ^{2}}{2}

定数を解答に追加する

= \frac{512 x ^{11}}{11} - \frac{1152 x ^{10}}{5} + 512 x^{9} - 672 x^{8} + 576 x^{7} - 336 x^{6} + \frac{672 x ^{5}}{5} - 36 x^{4} + 6 x^{3} - \frac{x ^{2}}{2} + C