integral of x(1-x)^7

解

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

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

解答ステップ

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

拡張

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

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

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

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

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

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

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

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

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

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

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

定数を解答に追加する

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