integral of (x^2+1)(x^3+3x-12)^4

解

\int (x^{2} + 1) (x^{3} + 3 x - 12)^{4} d x

\frac{x ^{15}}{15} + x^{13} - 4 x^{12} + 6 x^{11} - 48 x^{10} + 114 x^{9} - 216 x^{8} + 891 x^{7} - 1584 x^{6} + \frac{13041 x ^{5}}{5} - 7236 x^{4} + 9504 x^{3} - 10368 x^{2} + 20736 x + C

解答ステップ

\int (x^{2} + 1) (x^{3} + 3 x - 12)^{4} d x

拡張

(x^{2} + 1) (x^{3} + 3 x - 12)^{4} : x^{14} + 13 x^{12} - 48 x^{11} + 66 x^{10} - 480 x^{9} + 1026 x^{8} - 1728 x^{7} + 6237 x^{6} - 9504 x^{5} + 13041 x^{4} - 28944 x^{3} + 28512 x^{2} - 20736 x + 20736

= \int x^{14} + 13 x^{12} - 48 x^{11} + 66 x^{10} - 480 x^{9} + 1026 x^{8} - 1728 x^{7} + 6237 x^{6} - 9504 x^{5} + 13041 x^{4} - 28944 x^{3} + 28512 x^{2} - 20736 x + 20736 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^{14} d x + \int 13 x^{12} d x - \int 48 x^{11} d x + \int 66 x^{10} d x - \int 480 x^{9} d x + \int 1026 x^{8} d x - \int 1728 x^{7} d x + \int 6237 x^{6} d x - \int 9504 x^{5} d x + \int 13041 x^{4} d x - \int 28944 x^{3} d x + \int 28512 x^{2} d x - \int 20736 x d x + \int 20736 d x

\int x^{14} d x = \frac{x ^{15}}{15}

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

\int 48 x^{11} d x = 4 x^{12}

\int 66 x^{10} d x = 6 x^{11}

\int 480 x^{9} d x = 48 x^{10}

\int 1026 x^{8} d x = 114 x^{9}

\int 1728 x^{7} d x = 216 x^{8}

\int 6237 x^{6} d x = 891 x^{7}

\int 9504 x^{5} d x = 1584 x^{6}

\int 13041 x^{4} d x = \frac{13041 x ^{5}}{5}

\int 28944 x^{3} d x = 7236 x^{4}

\int 28512 x^{2} d x = 9504 x^{3}

\int 20736 x d x = 10368 x^{2}

\int 20736 d x = 20736 x

= \frac{x ^{15}}{15} + x^{13} - 4 x^{12} + 6 x^{11} - 48 x^{10} + 114 x^{9} - 216 x^{8} + 891 x^{7} - 1584 x^{6} + \frac{13041 x ^{5}}{5} - 7236 x^{4} + 9504 x^{3} - 10368 x^{2} + 20736 x

定数を解答に追加する

= \frac{x ^{15}}{15} + x^{13} - 4 x^{12} + 6 x^{11} - 48 x^{10} + 114 x^{9} - 216 x^{8} + 891 x^{7} - 1584 x^{6} + \frac{13041 x ^{5}}{5} - 7236 x^{4} + 9504 x^{3} - 10368 x^{2} + 20736 x + C