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

解答

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

\frac{x ^{33}}{33} + \frac{10 x ^{30}}{3} + \frac{500 x ^{27}}{3} + 5000 x^{24} + 100000 x^{21} + 1400000 x^{18} + 14000000 x^{15} + 100000000 x^{12} + 500000000 x^{9} + \frac{5000000000 x ^{6}}{3} + \frac{1 0 ^{10} x ^{3}}{3} + C

求解步骤

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

乘开

x^{2} (x^{3} + 10)^{10} : x^{32} + 100 x^{29} + 4500 x^{26} + 120000 x^{23} + 2100000 x^{20} + 25200000 x^{17} + 210000000 x^{14} + 1200000000 x^{11} + 4500000000 x^{8} + 1 0^{10} x^{5} + 1 0^{10} x^{2}

= \int x^{32} + 100 x^{29} + 4500 x^{26} + 120000 x^{23} + 2100000 x^{20} + 25200000 x^{17} + 210000000 x^{14} + 1200000000 x^{11} + 4500000000 x^{8} + 1 0^{10} x^{5} + 1 0^{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 100 x^{29} d x + \int 4500 x^{26} d x + \int 120000 x^{23} d x + \int 2100000 x^{20} d x + \int 25200000 x^{17} d x + \int 210000000 x^{14} d x + \int 1200000000 x^{11} d x + \int 4500000000 x^{8} d x + \int 1 0^{10} x^{5} d x + \int 1 0^{10} x^{2} d x

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

\int 100 x^{29} d x = \frac{10 x ^{30}}{3}

\int 4500 x^{26} d x = \frac{500 x ^{27}}{3}

\int 120000 x^{23} d x = 5000 x^{24}

\int 2100000 x^{20} d x = 100000 x^{21}

\int 25200000 x^{17} d x = 1400000 x^{18}

\int 210000000 x^{14} d x = 14000000 x^{15}

\int 1200000000 x^{11} d x = 100000000 x^{12}

\int 4500000000 x^{8} d x = 500000000 x^{9}

\int 1 0^{10} x^{5} d x = \frac{5000000000 x ^{6}}{3}

\int 1 0^{10} x^{2} d x = \frac{1 0 ^{10} x ^{3}}{3}

= \frac{x ^{33}}{33} + \frac{10 x ^{30}}{3} + \frac{500 x ^{27}}{3} + 5000 x^{24} + 100000 x^{21} + 1400000 x^{18} + 14000000 x^{15} + 100000000 x^{12} + 500000000 x^{9} + \frac{5000000000 x ^{6}}{3} + \frac{1 0 ^{10} x ^{3}}{3}

解答补常数

= \frac{x ^{33}}{33} + \frac{10 x ^{30}}{3} + \frac{500 x ^{27}}{3} + 5000 x^{24} + 100000 x^{21} + 1400000 x^{18} + 14000000 x^{15} + 100000000 x^{12} + 500000000 x^{9} + \frac{5000000000 x ^{6}}{3} + \frac{1 0 ^{10} x ^{3}}{3} + C

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