integral of (x^{20}-5x^4)^6(x^{19}-x^3)

解

\int (x^{20} - 5 x^{4})^{6} (x^{19} - x^{3}) d x

\frac{x ^{140}}{140} - \frac{x ^{124}}{4} + \frac{15 x ^{108}}{4} - \frac{125 x ^{92}}{4} + \frac{625 x ^{76}}{4} - \frac{1875 x ^{60}}{4} + \frac{3125 x ^{44}}{4} - \frac{15625 x ^{28}}{28} + C

解答ステップ

\int (x^{20} - 5 x^{4})^{6} (x^{19} - x^{3}) d x

拡張

(x^{20} - 5 x^{4})^{6} (x^{19} - x^{3}) : x^{139} - 31 x^{123} + 405 x^{107} - 2875 x^{91} + 11875 x^{75} - 28125 x^{59} + 34375 x^{43} - 15625 x^{27}

= \int x^{139} - 31 x^{123} + 405 x^{107} - 2875 x^{91} + 11875 x^{75} - 28125 x^{59} + 34375 x^{43} - 15625 x^{27} 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^{139} d x - \int 31 x^{123} d x + \int 405 x^{107} d x - \int 2875 x^{91} d x + \int 11875 x^{75} d x - \int 28125 x^{59} d x + \int 34375 x^{43} d x - \int 15625 x^{27} d x

\int x^{139} d x = \frac{x ^{140}}{140}

\int 31 x^{123} d x = \frac{x ^{124}}{4}

\int 405 x^{107} d x = \frac{15 x ^{108}}{4}

\int 2875 x^{91} d x = \frac{125 x ^{92}}{4}

\int 11875 x^{75} d x = \frac{625 x ^{76}}{4}

\int 28125 x^{59} d x = \frac{1875 x ^{60}}{4}

\int 34375 x^{43} d x = \frac{3125 x ^{44}}{4}

\int 15625 x^{27} d x = \frac{15625 x ^{28}}{28}

= \frac{x ^{140}}{140} - \frac{x ^{124}}{4} + \frac{15 x ^{108}}{4} - \frac{125 x ^{92}}{4} + \frac{625 x ^{76}}{4} - \frac{1875 x ^{60}}{4} + \frac{3125 x ^{44}}{4} - \frac{15625 x ^{28}}{28}

定数を解答に追加する

= \frac{x ^{140}}{140} - \frac{x ^{124}}{4} + \frac{15 x ^{108}}{4} - \frac{125 x ^{92}}{4} + \frac{625 x ^{76}}{4} - \frac{1875 x ^{60}}{4} + \frac{3125 x ^{44}}{4} - \frac{15625 x ^{28}}{28} + C