integral of x^2(x^3+7)^{10}

解答

\int x^{2} (x^{3} + 7)^{10} d x

\frac{x ^{33}}{33} + \frac{7 x ^{30}}{3} + \frac{245 x ^{27}}{3} + 1715 x^{24} + 24010 x^{21} + 235298 x^{18} + 1647086 x^{15} + 8235430 x^{12} + 28824005 x^{9} + \frac{201768035 x ^{6}}{3} + \frac{282475249 x ^{3}}{3} + C

求解步骤

\int x^{2} (x^{3} + 7)^{10} d x

乘开

x^{2} (x^{3} + 7)^{10} : x^{32} + 70 x^{29} + 2205 x^{26} + 41160 x^{23} + 504210 x^{20} + 4235364 x^{17} + 24706290 x^{14} + 98825160 x^{11} + 259416045 x^{8} + 403536070 x^{5} + 282475249 x^{2}

= \int x^{32} + 70 x^{29} + 2205 x^{26} + 41160 x^{23} + 504210 x^{20} + 4235364 x^{17} + 24706290 x^{14} + 98825160 x^{11} + 259416045 x^{8} + 403536070 x^{5} + 282475249 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 70 x^{29} d x + \int 2205 x^{26} d x + \int 41160 x^{23} d x + \int 504210 x^{20} d x + \int 4235364 x^{17} d x + \int 24706290 x^{14} d x + \int 98825160 x^{11} d x + \int 259416045 x^{8} d x + \int 403536070 x^{5} d x + \int 282475249 x^{2} d x

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

\int 70 x^{29} d x = \frac{7 x ^{30}}{3}

\int 2205 x^{26} d x = \frac{245 x ^{27}}{3}

\int 41160 x^{23} d x = 1715 x^{24}

\int 504210 x^{20} d x = 24010 x^{21}

\int 4235364 x^{17} d x = 235298 x^{18}

\int 24706290 x^{14} d x = 1647086 x^{15}

\int 98825160 x^{11} d x = 8235430 x^{12}

\int 259416045 x^{8} d x = 28824005 x^{9}

\int 403536070 x^{5} d x = \frac{201768035 x ^{6}}{3}

\int 282475249 x^{2} d x = \frac{282475249 x ^{3}}{3}

= \frac{x ^{33}}{33} + \frac{7 x ^{30}}{3} + \frac{245 x ^{27}}{3} + 1715 x^{24} + 24010 x^{21} + 235298 x^{18} + 1647086 x^{15} + 8235430 x^{12} + 28824005 x^{9} + \frac{201768035 x ^{6}}{3} + \frac{282475249 x ^{3}}{3}

解答补常数

= \frac{x ^{33}}{33} + \frac{7 x ^{30}}{3} + \frac{245 x ^{27}}{3} + 1715 x^{24} + 24010 x^{21} + 235298 x^{18} + 1647086 x^{15} + 8235430 x^{12} + 28824005 x^{9} + \frac{201768035 x ^{6}}{3} + \frac{282475249 x ^{3}}{3} + C

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