integral of x^2(x^3+9)^{10}

解答

\int x^{2} (x^{3} + 9)^{10} d x

\frac{x ^{33}}{33} + 3 x^{30} + 135 x^{27} + 3645 x^{24} + 65610 x^{21} + 826686 x^{18} + 7440174 x^{15} + 47829690 x^{12} + 215233605 x^{9} + 645700815 x^{6} + 3^{19} x^{3} + C

求解步骤

\int x^{2} (x^{3} + 9)^{10} d x

乘开

x^{2} (x^{3} + 9)^{10} : x^{32} + 90 x^{29} + 3645 x^{26} + 87480 x^{23} + 1377810 x^{20} + 14880348 x^{17} + 111602610 x^{14} + 573956280 x^{11} + 1937102445 x^{8} + 3874204890 x^{5} + 9^{10} x^{2}

= \int x^{32} + 90 x^{29} + 3645 x^{26} + 87480 x^{23} + 1377810 x^{20} + 14880348 x^{17} + 111602610 x^{14} + 573956280 x^{11} + 1937102445 x^{8} + 3874204890 x^{5} + 9^{10} 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 90 x^{29} d x + \int 3645 x^{26} d x + \int 87480 x^{23} d x + \int 1377810 x^{20} d x + \int 14880348 x^{17} d x + \int 111602610 x^{14} d x + \int 573956280 x^{11} d x + \int 1937102445 x^{8} d x + \int 3874204890 x^{5} d x + \int 9^{10} x^{2} d x

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

\int 90 x^{29} d x = 3 x^{30}

\int 3645 x^{26} d x = 135 x^{27}

\int 87480 x^{23} d x = 3645 x^{24}

\int 1377810 x^{20} d x = 65610 x^{21}

\int 14880348 x^{17} d x = 826686 x^{18}

\int 111602610 x^{14} d x = 7440174 x^{15}

\int 573956280 x^{11} d x = 47829690 x^{12}

\int 1937102445 x^{8} d x = 215233605 x^{9}

\int 3874204890 x^{5} d x = 645700815 x^{6}

\int 9^{10} x^{2} d x = 3^{19} x^{3}

= \frac{x ^{33}}{33} + 3 x^{30} + 135 x^{27} + 3645 x^{24} + 65610 x^{21} + 826686 x^{18} + 7440174 x^{15} + 47829690 x^{12} + 215233605 x^{9} + 645700815 x^{6} + 3^{19} x^{3}

解答补常数

= \frac{x ^{33}}{33} + 3 x^{30} + 135 x^{27} + 3645 x^{24} + 65610 x^{21} + 826686 x^{18} + 7440174 x^{15} + 47829690 x^{12} + 215233605 x^{9} + 645700815 x^{6} + 3^{19} x^{3} + C

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