integral of x(x+2)^9

解

\int x (x + 2)^{9} d x

\frac{x ^{11}}{11} + \frac{9 x ^{10}}{5} + 16 x^{9} + 84 x^{8} + 288 x^{7} + 672 x^{6} + \frac{5376 x ^{5}}{5} + 1152 x^{4} + 768 x^{3} + 256 x^{2} + C

解答ステップ

\int x (x + 2)^{9} d x

拡張

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

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

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

\int 18 x^{9} d x = \frac{9 x ^{10}}{5}

\int 144 x^{8} d x = 16 x^{9}

\int 672 x^{7} d x = 84 x^{8}

\int 2016 x^{6} d x = 288 x^{7}

\int 4032 x^{5} d x = 672 x^{6}

\int 5376 x^{4} d x = \frac{5376 x ^{5}}{5}

\int 4608 x^{3} d x = 1152 x^{4}

\int 2304 x^{2} d x = 768 x^{3}

\int 512 x d x = 256 x^{2}

= \frac{x ^{11}}{11} + \frac{9 x ^{10}}{5} + 16 x^{9} + 84 x^{8} + 288 x^{7} + 672 x^{6} + \frac{5376 x ^{5}}{5} + 1152 x^{4} + 768 x^{3} + 256 x^{2}

定数を解答に追加する

= \frac{x ^{11}}{11} + \frac{9 x ^{10}}{5} + 16 x^{9} + 84 x^{8} + 288 x^{7} + 672 x^{6} + \frac{5376 x ^{5}}{5} + 1152 x^{4} + 768 x^{3} + 256 x^{2} + C