integral of x(x-1)^7

解

\int x (x - 1)^{7} d x

\frac{x ^{9}}{9} - \frac{7 x ^{8}}{8} + 3 x^{7} - \frac{35 x ^{6}}{6} + 7 x^{5} - \frac{21 x ^{4}}{4} + \frac{7 x ^{3}}{3} - \frac{x ^{2}}{2} + C

解答ステップ

\int x (x - 1)^{7} d x

拡張

x (x - 1)^{7} : x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{3} + 7 x^{2} - x

= \int x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{3} + 7 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^{8} d x - \int 7 x^{7} d x + \int 21 x^{6} d x - \int 35 x^{5} d x + \int 35 x^{4} d x - \int 21 x^{3} d x + \int 7 x^{2} d x - \int x d x

\int x^{8} d x = \frac{x ^{9}}{9}

\int 7 x^{7} d x = \frac{7 x ^{8}}{8}

\int 21 x^{6} d x = 3 x^{7}

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

\int 35 x^{4} d x = 7 x^{5}

\int 21 x^{3} d x = \frac{21 x ^{4}}{4}

\int 7 x^{2} d x = \frac{7 x ^{3}}{3}

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

= \frac{x ^{9}}{9} - \frac{7 x ^{8}}{8} + 3 x^{7} - \frac{35 x ^{6}}{6} + 7 x^{5} - \frac{21 x ^{4}}{4} + \frac{7 x ^{3}}{3} - \frac{x ^{2}}{2}

定数を解答に追加する

= \frac{x ^{9}}{9} - \frac{7 x ^{8}}{8} + 3 x^{7} - \frac{35 x ^{6}}{6} + 7 x^{5} - \frac{21 x ^{4}}{4} + \frac{7 x ^{3}}{3} - \frac{x ^{2}}{2} + C