integral of x(x^2-7)^{10}

解答

\int x (x^{2} - 7)^{10} d x

\frac{x ^{22}}{22} - \frac{7 x ^{20}}{2} + \frac{245 x ^{18}}{2} - \frac{5145 x ^{16}}{2} + 36015 x^{14} - 352947 x^{12} + 2470629 x^{10} - 12353145 x^{8} + \frac{86472015 x ^{6}}{2} - \frac{201768035 x ^{4}}{2} + \frac{282475249 x ^{2}}{2} + C

求解步骤

\int x (x^{2} - 7)^{10} d x

乘开

x (x^{2} - 7)^{10} : x^{21} - 70 x^{19} + 2205 x^{17} - 41160 x^{15} + 504210 x^{13} - 4235364 x^{11} + 24706290 x^{9} - 98825160 x^{7} + 259416045 x^{5} - 403536070 x^{3} + 282475249 x

= \int x^{21} - 70 x^{19} + 2205 x^{17} - 41160 x^{15} + 504210 x^{13} - 4235364 x^{11} + 24706290 x^{9} - 98825160 x^{7} + 259416045 x^{5} - 403536070 x^{3} + 282475249 x 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^{21} d x - \int 70 x^{19} d x + \int 2205 x^{17} d x - \int 41160 x^{15} d x + \int 504210 x^{13} d x - \int 4235364 x^{11} d x + \int 24706290 x^{9} d x - \int 98825160 x^{7} d x + \int 259416045 x^{5} d x - \int 403536070 x^{3} d x + \int 282475249 x d x

\int x^{21} d x = \frac{x ^{22}}{22}

\int 70 x^{19} d x = \frac{7 x ^{20}}{2}

\int 2205 x^{17} d x = \frac{245 x ^{18}}{2}

\int 41160 x^{15} d x = \frac{5145 x ^{16}}{2}

\int 504210 x^{13} d x = 36015 x^{14}

\int 4235364 x^{11} d x = 352947 x^{12}

\int 24706290 x^{9} d x = 2470629 x^{10}

\int 98825160 x^{7} d x = 12353145 x^{8}

\int 259416045 x^{5} d x = \frac{86472015 x ^{6}}{2}

\int 403536070 x^{3} d x = \frac{201768035 x ^{4}}{2}

\int 282475249 x d x = \frac{282475249 x ^{2}}{2}

= \frac{x ^{22}}{22} - \frac{7 x ^{20}}{2} + \frac{245 x ^{18}}{2} - \frac{5145 x ^{16}}{2} + 36015 x^{14} - 352947 x^{12} + 2470629 x^{10} - 12353145 x^{8} + \frac{86472015 x ^{6}}{2} - \frac{201768035 x ^{4}}{2} + \frac{282475249 x ^{2}}{2}

解答补常数

= \frac{x ^{22}}{22} - \frac{7 x ^{20}}{2} + \frac{245 x ^{18}}{2} - \frac{5145 x ^{16}}{2} + 36015 x^{14} - 352947 x^{12} + 2470629 x^{10} - 12353145 x^{8} + \frac{86472015 x ^{6}}{2} - \frac{201768035 x ^{4}}{2} + \frac{282475249 x ^{2}}{2} + C

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