integral of x^2(x^3+6)^{10}

解答

\int x^{2} (x^{3} + 6)^{10} d x

\frac{x ^{33}}{33} + 2 x^{30} + 60 x^{27} + 1080 x^{24} + 12960 x^{21} + 108864 x^{18} + 653184 x^{15} + 2799360 x^{12} + 8398080 x^{9} + 16796160 x^{6} + 20155392 x^{3} + C

求解步骤

\int x^{2} (x^{3} + 6)^{10} d x

乘开

x^{2} (x^{3} + 6)^{10} : x^{32} + 60 x^{29} + 1620 x^{26} + 25920 x^{23} + 272160 x^{20} + 1959552 x^{17} + 9797760 x^{14} + 33592320 x^{11} + 75582720 x^{8} + 100776960 x^{5} + 60466176 x^{2}

= \int x^{32} + 60 x^{29} + 1620 x^{26} + 25920 x^{23} + 272160 x^{20} + 1959552 x^{17} + 9797760 x^{14} + 33592320 x^{11} + 75582720 x^{8} + 100776960 x^{5} + 60466176 x^{2} 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 x^{32} d x + \int 60 x^{29} d x + \int 1620 x^{26} d x + \int 25920 x^{23} d x + \int 272160 x^{20} d x + \int 1959552 x^{17} d x + \int 9797760 x^{14} d x + \int 33592320 x^{11} d x + \int 75582720 x^{8} d x + \int 100776960 x^{5} d x + \int 60466176 x^{2} d x

\int x^{32} d x = \frac{x ^{33}}{33}

\int 60 x^{29} d x = 2 x^{30}

\int 1620 x^{26} d x = 60 x^{27}

\int 25920 x^{23} d x = 1080 x^{24}

\int 272160 x^{20} d x = 12960 x^{21}

\int 1959552 x^{17} d x = 108864 x^{18}

\int 9797760 x^{14} d x = 653184 x^{15}

\int 33592320 x^{11} d x = 2799360 x^{12}

\int 75582720 x^{8} d x = 8398080 x^{9}

\int 100776960 x^{5} d x = 16796160 x^{6}

\int 60466176 x^{2} d x = 20155392 x^{3}

= \frac{x ^{33}}{33} + 2 x^{30} + 60 x^{27} + 1080 x^{24} + 12960 x^{21} + 108864 x^{18} + 653184 x^{15} + 2799360 x^{12} + 8398080 x^{9} + 16796160 x^{6} + 20155392 x^{3}

解答补常数

= \frac{x ^{33}}{33} + 2 x^{30} + 60 x^{27} + 1080 x^{24} + 12960 x^{21} + 108864 x^{18} + 653184 x^{15} + 2799360 x^{12} + 8398080 x^{9} + 16796160 x^{6} + 20155392 x^{3} + C

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