integral of (x^3-7x)^4(3x^2-7)

解

\int (x^{3} - 7 x)^{4} (3 x^{2} - 7) d x

\frac{x ^{15}}{5} - 7 x^{13} + 98 x^{11} - 686 x^{9} + 2401 x^{7} - \frac{16807 x ^{5}}{5} + C

解答ステップ

\int (x^{3} - 7 x)^{4} (3 x^{2} - 7) d x

拡張

(x^{3} - 7 x)^{4} (3 x^{2} - 7) : 3 x^{14} - 91 x^{12} + 1078 x^{10} - 6174 x^{8} + 16807 x^{6} - 16807 x^{4}

= \int 3 x^{14} - 91 x^{12} + 1078 x^{10} - 6174 x^{8} + 16807 x^{6} - 16807 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 3 x^{14} d x - \int 91 x^{12} d x + \int 1078 x^{10} d x - \int 6174 x^{8} d x + \int 16807 x^{6} d x - \int 16807 x^{4} d x

\int 3 x^{14} d x = \frac{x ^{15}}{5}

\int 91 x^{12} d x = 7 x^{13}

\int 1078 x^{10} d x = 98 x^{11}

\int 6174 x^{8} d x = 686 x^{9}

\int 16807 x^{6} d x = 2401 x^{7}

\int 16807 x^{4} d x = \frac{16807 x ^{5}}{5}

= \frac{x ^{15}}{5} - 7 x^{13} + 98 x^{11} - 686 x^{9} + 2401 x^{7} - \frac{16807 x ^{5}}{5}

定数を解答に追加する

= \frac{x ^{15}}{5} - 7 x^{13} + 98 x^{11} - 686 x^{9} + 2401 x^{7} - \frac{16807 x ^{5}}{5} + C