integral of (3x^2+2x)^9(6x+2)

解答

\int (3 x^{2} + 2 x)^{9} (6 x + 2) d x

\frac{59049 x ^{20}}{10} + 39366 x^{19} + 118098 x^{18} + 209952 x^{17} + 244944 x^{16} + \frac{979776 x ^{15}}{5} + 108864 x^{14} + 41472 x^{13} + 10368 x^{12} + 1536 x^{11} + \frac{512 x ^{10}}{5} + C

求解步骤

\int (3 x^{2} + 2 x)^{9} (6 x + 2) d x

乘开

(3 x^{2} + 2 x)^{9} (6 x + 2) : 118098 x^{19} + 747954 x^{18} + 2125764 x^{17} + 3569184 x^{16} + 3919104 x^{15} + 2939328 x^{14} + 1524096 x^{13} + 539136 x^{12} + 124416 x^{11} + 16896 x^{10} + 1024 x^{9}

= \int 118098 x^{19} + 747954 x^{18} + 2125764 x^{17} + 3569184 x^{16} + 3919104 x^{15} + 2939328 x^{14} + 1524096 x^{13} + 539136 x^{12} + 124416 x^{11} + 16896 x^{10} + 1024 x^{9} 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 118098 x^{19} d x + \int 747954 x^{18} d x + \int 2125764 x^{17} d x + \int 3569184 x^{16} d x + \int 3919104 x^{15} d x + \int 2939328 x^{14} d x + \int 1524096 x^{13} d x + \int 539136 x^{12} d x + \int 124416 x^{11} d x + \int 16896 x^{10} d x + \int 1024 x^{9} d x

\int 118098 x^{19} d x = \frac{59049 x ^{20}}{10}

\int 747954 x^{18} d x = 39366 x^{19}

\int 2125764 x^{17} d x = 118098 x^{18}

\int 3569184 x^{16} d x = 209952 x^{17}

\int 3919104 x^{15} d x = 244944 x^{16}

\int 2939328 x^{14} d x = \frac{979776 x ^{15}}{5}

\int 1524096 x^{13} d x = 108864 x^{14}

\int 539136 x^{12} d x = 41472 x^{13}

\int 124416 x^{11} d x = 10368 x^{12}

\int 16896 x^{10} d x = 1536 x^{11}

\int 1024 x^{9} d x = \frac{512 x ^{10}}{5}

= \frac{59049 x ^{20}}{10} + 39366 x^{19} + 118098 x^{18} + 209952 x^{17} + 244944 x^{16} + \frac{979776 x ^{15}}{5} + 108864 x^{14} + 41472 x^{13} + 10368 x^{12} + 1536 x^{11} + \frac{512 x ^{10}}{5}

解答补常数

= \frac{59049 x ^{20}}{10} + 39366 x^{19} + 118098 x^{18} + 209952 x^{17} + 244944 x^{16} + \frac{979776 x ^{15}}{5} + 108864 x^{14} + 41472 x^{13} + 10368 x^{12} + 1536 x^{11} + \frac{512 x ^{10}}{5} + C

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