integral of (x^4-5x^2-6x)^4(4x^3-10x-6)

解答

\int (x^{4} - 5 x^{2} - 6 x)^{4} (4 x^{3} - 10 x - 6) d x

\frac{x ^{20}}{5} - 5 x^{18} - 6 x^{17} + 50 x^{16} + 120 x^{15} - 178 x^{14} - 900 x^{13} - 455 x^{12} + 2568 x^{11} + 4775 x^{10} + 570 x^{9} - 7704 x^{8} - 10800 x^{7} - 6480 x^{6} - \frac{7776 x ^{5}}{5} + C

求解步骤

\int (x^{4} - 5 x^{2} - 6 x)^{4} (4 x^{3} - 10 x - 6) d x

乘开

(x^{4} - 5 x^{2} - 6 x)^{4} (4 x^{3} - 10 x - 6) : 4 x^{19} - 90 x^{17} - 102 x^{16} + 800 x^{15} + 1800 x^{14} - 2492 x^{13} - 11700 x^{12} - 5460 x^{11} + 28248 x^{10} + 47750 x^{9} + 5130 x^{8} - 61632 x^{7} - 75600 x^{6} - 38880 x^{5} - 7776 x^{4}

= \int 4 x^{19} - 90 x^{17} - 102 x^{16} + 800 x^{15} + 1800 x^{14} - 2492 x^{13} - 11700 x^{12} - 5460 x^{11} + 28248 x^{10} + 47750 x^{9} + 5130 x^{8} - 61632 x^{7} - 75600 x^{6} - 38880 x^{5} - 7776 x^{4} d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 4 x^{19} d x - \int 90 x^{17} d x - \int 102 x^{16} d x + \int 800 x^{15} d x + \int 1800 x^{14} d x - \int 2492 x^{13} d x - \int 11700 x^{12} d x - \int 5460 x^{11} d x + \int 28248 x^{10} d x + \int 47750 x^{9} d x + \int 5130 x^{8} d x - \int 61632 x^{7} d x - \int 75600 x^{6} d x - \int 38880 x^{5} d x - \int 7776 x^{4} d x

\int 4 x^{19} d x = \frac{x ^{20}}{5}

\int 90 x^{17} d x = 5 x^{18}

\int 102 x^{16} d x = 6 x^{17}

\int 800 x^{15} d x = 50 x^{16}

\int 1800 x^{14} d x = 120 x^{15}

\int 2492 x^{13} d x = 178 x^{14}

\int 11700 x^{12} d x = 900 x^{13}

\int 5460 x^{11} d x = 455 x^{12}

\int 28248 x^{10} d x = 2568 x^{11}

\int 47750 x^{9} d x = 4775 x^{10}

\int 5130 x^{8} d x = 570 x^{9}

\int 61632 x^{7} d x = 7704 x^{8}

\int 75600 x^{6} d x = 10800 x^{7}

\int 38880 x^{5} d x = 6480 x^{6}

\int 7776 x^{4} d x = \frac{7776 x ^{5}}{5}

= \frac{x ^{20}}{5} - 5 x^{18} - 6 x^{17} + 50 x^{16} + 120 x^{15} - 178 x^{14} - 900 x^{13} - 455 x^{12} + 2568 x^{11} + 4775 x^{10} + 570 x^{9} - 7704 x^{8} - 10800 x^{7} - 6480 x^{6} - \frac{7776 x ^{5}}{5}

解答补常数

= \frac{x ^{20}}{5} - 5 x^{18} - 6 x^{17} + 50 x^{16} + 120 x^{15} - 178 x^{14} - 900 x^{13} - 455 x^{12} + 2568 x^{11} + 4775 x^{10} + 570 x^{9} - 7704 x^{8} - 10800 x^{7} - 6480 x^{6} - \frac{7776 x ^{5}}{5} + C

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