integral of x^7(x^4+1)^{10}

解答

\int x^{7} (x^{4} + 1)^{10} d x

\frac{x ^{48}}{48} + \frac{5 x ^{44}}{22} + \frac{9 x ^{40}}{8} + \frac{10 x ^{36}}{3} + \frac{105 x ^{32}}{16} + 9 x^{28} + \frac{35 x ^{24}}{4} + 6 x^{20} + \frac{45 x ^{16}}{16} + \frac{5 x ^{12}}{6} + \frac{x ^{8}}{8} + C

求解步骤

\int x^{7} (x^{4} + 1)^{10} d x

乘开

x^{7} (x^{4} + 1)^{10} : x^{47} + 10 x^{43} + 45 x^{39} + 120 x^{35} + 210 x^{31} + 252 x^{27} + 210 x^{23} + 120 x^{19} + 45 x^{15} + 10 x^{11} + x^{7}

= \int x^{47} + 10 x^{43} + 45 x^{39} + 120 x^{35} + 210 x^{31} + 252 x^{27} + 210 x^{23} + 120 x^{19} + 45 x^{15} + 10 x^{11} + x^{7} 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^{47} d x + \int 10 x^{43} d x + \int 45 x^{39} d x + \int 120 x^{35} d x + \int 210 x^{31} d x + \int 252 x^{27} d x + \int 210 x^{23} d x + \int 120 x^{19} d x + \int 45 x^{15} d x + \int 10 x^{11} d x + \int x^{7} d x

\int x^{47} d x = \frac{x ^{48}}{48}

\int 10 x^{43} d x = \frac{5 x ^{44}}{22}

\int 45 x^{39} d x = \frac{9 x ^{40}}{8}

\int 120 x^{35} d x = \frac{10 x ^{36}}{3}

\int 210 x^{31} d x = \frac{105 x ^{32}}{16}

\int 252 x^{27} d x = 9 x^{28}

\int 210 x^{23} d x = \frac{35 x ^{24}}{4}

\int 120 x^{19} d x = 6 x^{20}

\int 45 x^{15} d x = \frac{45 x ^{16}}{16}

\int 10 x^{11} d x = \frac{5 x ^{12}}{6}

\int x^{7} d x = \frac{x ^{8}}{8}

= \frac{x ^{48}}{48} + \frac{5 x ^{44}}{22} + \frac{9 x ^{40}}{8} + \frac{10 x ^{36}}{3} + \frac{105 x ^{32}}{16} + 9 x^{28} + \frac{35 x ^{24}}{4} + 6 x^{20} + \frac{45 x ^{16}}{16} + \frac{5 x ^{12}}{6} + \frac{x ^{8}}{8}

解答补常数

= \frac{x ^{48}}{48} + \frac{5 x ^{44}}{22} + \frac{9 x ^{40}}{8} + \frac{10 x ^{36}}{3} + \frac{105 x ^{32}}{16} + 9 x^{28} + \frac{35 x ^{24}}{4} + 6 x^{20} + \frac{45 x ^{16}}{16} + \frac{5 x ^{12}}{6} + \frac{x ^{8}}{8} + C

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