integral of (x-1)(x^2-2x+5)^6

解

\int (x - 1) (x^{2} - 2 x + 5)^{6} d x

\frac{x ^{14}}{14} - x^{13} + \frac{17 x ^{12}}{2} - 50 x^{11} + \frac{455 x ^{10}}{2} - 823 x^{9} + \frac{4889 x ^{8}}{2} - \frac{41924 x ^{7}}{7} + \frac{24445 x ^{6}}{2} - 20575 x^{5} + \frac{56875 x ^{4}}{2} - 31250 x^{3} + \frac{53125 x ^{2}}{2} - 15625 x + C

解答ステップ

\int (x - 1) (x^{2} - 2 x + 5)^{6} d x

拡張

(x - 1) (x^{2} - 2 x + 5)^{6} : x^{13} - 13 x^{12} + 102 x^{11} - 550 x^{10} + 2275 x^{9} - 7407 x^{8} + 19556 x^{7} - 41924 x^{6} + 73335 x^{5} - 102875 x^{4} + 113750 x^{3} - 93750 x^{2} + 53125 x - 15625

= \int x^{13} - 13 x^{12} + 102 x^{11} - 550 x^{10} + 2275 x^{9} - 7407 x^{8} + 19556 x^{7} - 41924 x^{6} + 73335 x^{5} - 102875 x^{4} + 113750 x^{3} - 93750 x^{2} + 53125 x - 15625 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^{13} d x - \int 13 x^{12} d x + \int 102 x^{11} d x - \int 550 x^{10} d x + \int 2275 x^{9} d x - \int 7407 x^{8} d x + \int 19556 x^{7} d x - \int 41924 x^{6} d x + \int 73335 x^{5} d x - \int 102875 x^{4} d x + \int 113750 x^{3} d x - \int 93750 x^{2} d x + \int 53125 x d x - \int 15625 d x

\int x^{13} d x = \frac{x ^{14}}{14}

\int 13 x^{12} d x = x^{13}

\int 102 x^{11} d x = \frac{17 x ^{12}}{2}

\int 550 x^{10} d x = 50 x^{11}

\int 2275 x^{9} d x = \frac{455 x ^{10}}{2}

\int 7407 x^{8} d x = 823 x^{9}

\int 19556 x^{7} d x = \frac{4889 x ^{8}}{2}

\int 41924 x^{6} d x = \frac{41924 x ^{7}}{7}

\int 73335 x^{5} d x = \frac{24445 x ^{6}}{2}

\int 102875 x^{4} d x = 20575 x^{5}

\int 113750 x^{3} d x = \frac{56875 x ^{4}}{2}

\int 93750 x^{2} d x = 31250 x^{3}

\int 53125 x d x = \frac{53125 x ^{2}}{2}

\int 15625 d x = 15625 x

= \frac{x ^{14}}{14} - x^{13} + \frac{17 x ^{12}}{2} - 50 x^{11} + \frac{455 x ^{10}}{2} - 823 x^{9} + \frac{4889 x ^{8}}{2} - \frac{41924 x ^{7}}{7} + \frac{24445 x ^{6}}{2} - 20575 x^{5} + \frac{56875 x ^{4}}{2} - 31250 x^{3} + \frac{53125 x ^{2}}{2} - 15625 x

定数を解答に追加する

= \frac{x ^{14}}{14} - x^{13} + \frac{17 x ^{12}}{2} - 50 x^{11} + \frac{455 x ^{10}}{2} - 823 x^{9} + \frac{4889 x ^{8}}{2} - \frac{41924 x ^{7}}{7} + \frac{24445 x ^{6}}{2} - 20575 x^{5} + \frac{56875 x ^{4}}{2} - 31250 x^{3} + \frac{53125 x ^{2}}{2} - 15625 x + C