integral of x^4(x^5-1)^7

解

\int x^{4} (x^{5} - 1)^{7} d x

\frac{x ^{40}}{40} - \frac{x ^{35}}{5} + \frac{7 x ^{30}}{10} - \frac{7 x ^{25}}{5} + \frac{7 x ^{20}}{4} - \frac{7 x ^{15}}{5} + \frac{7 x ^{10}}{10} - \frac{x ^{5}}{5} + C

解答ステップ

\int x^{4} (x^{5} - 1)^{7} d x

拡張

x^{4} (x^{5} - 1)^{7} : x^{39} - 7 x^{34} + 21 x^{29} - 35 x^{24} + 35 x^{19} - 21 x^{14} + 7 x^{9} - x^{4}

= \int x^{39} - 7 x^{34} + 21 x^{29} - 35 x^{24} + 35 x^{19} - 21 x^{14} + 7 x^{9} - x^{4} 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^{39} d x - \int 7 x^{34} d x + \int 21 x^{29} d x - \int 35 x^{24} d x + \int 35 x^{19} d x - \int 21 x^{14} d x + \int 7 x^{9} d x - \int x^{4} d x

\int x^{39} d x = \frac{x ^{40}}{40}

\int 7 x^{34} d x = \frac{x ^{35}}{5}

\int 21 x^{29} d x = \frac{7 x ^{30}}{10}

\int 35 x^{24} d x = \frac{7 x ^{25}}{5}

\int 35 x^{19} d x = \frac{7 x ^{20}}{4}

\int 21 x^{14} d x = \frac{7 x ^{15}}{5}

\int 7 x^{9} d x = \frac{7 x ^{10}}{10}

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

= \frac{x ^{40}}{40} - \frac{x ^{35}}{5} + \frac{7 x ^{30}}{10} - \frac{7 x ^{25}}{5} + \frac{7 x ^{20}}{4} - \frac{7 x ^{15}}{5} + \frac{7 x ^{10}}{10} - \frac{x ^{5}}{5}

定数を解答に追加する

= \frac{x ^{40}}{40} - \frac{x ^{35}}{5} + \frac{7 x ^{30}}{10} - \frac{7 x ^{25}}{5} + \frac{7 x ^{20}}{4} - \frac{7 x ^{15}}{5} + \frac{7 x ^{10}}{10} - \frac{x ^{5}}{5} + C