integral of x(x-1)^{10}

解

\int x (x - 1)^{10} d x

\frac{x ^{12}}{12} - \frac{10 x ^{11}}{11} + \frac{9 x ^{10}}{2} - \frac{40 x ^{9}}{3} + \frac{105 x ^{8}}{4} - 36 x^{7} + 35 x^{6} - 24 x^{5} + \frac{45 x ^{4}}{4} - \frac{10 x ^{3}}{3} + \frac{x ^{2}}{2} + C

解答ステップ

\int x (x - 1)^{10} d x

拡張

x (x - 1)^{10} : x^{11} - 10 x^{10} + 45 x^{9} - 120 x^{8} + 210 x^{7} - 252 x^{6} + 210 x^{5} - 120 x^{4} + 45 x^{3} - 10 x^{2} + x

= \int x^{11} - 10 x^{10} + 45 x^{9} - 120 x^{8} + 210 x^{7} - 252 x^{6} + 210 x^{5} - 120 x^{4} + 45 x^{3} - 10 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 x^{11} d x - \int 10 x^{10} d x + \int 45 x^{9} d x - \int 120 x^{8} d x + \int 210 x^{7} d x - \int 252 x^{6} d x + \int 210 x^{5} d x - \int 120 x^{4} d x + \int 45 x^{3} d x - \int 10 x^{2} d x + \int x d x

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

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

\int 45 x^{9} d x = \frac{9 x ^{10}}{2}

\int 120 x^{8} d x = \frac{40 x ^{9}}{3}

\int 210 x^{7} d x = \frac{105 x ^{8}}{4}

\int 252 x^{6} d x = 36 x^{7}

\int 210 x^{5} d x = 35 x^{6}

\int 120 x^{4} d x = 24 x^{5}

\int 45 x^{3} d x = \frac{45 x ^{4}}{4}

\int 10 x^{2} d x = \frac{10 x ^{3}}{3}

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

= \frac{x ^{12}}{12} - \frac{10 x ^{11}}{11} + \frac{9 x ^{10}}{2} - \frac{40 x ^{9}}{3} + \frac{105 x ^{8}}{4} - 36 x^{7} + 35 x^{6} - 24 x^{5} + \frac{45 x ^{4}}{4} - \frac{10 x ^{3}}{3} + \frac{x ^{2}}{2}

定数を解答に追加する

= \frac{x ^{12}}{12} - \frac{10 x ^{11}}{11} + \frac{9 x ^{10}}{2} - \frac{40 x ^{9}}{3} + \frac{105 x ^{8}}{4} - 36 x^{7} + 35 x^{6} - 24 x^{5} + \frac{45 x ^{4}}{4} - \frac{10 x ^{3}}{3} + \frac{x ^{2}}{2} + C