integral of (5x^8-11)^7(40x^7)

解答

\int (5 x^{8} - 11)^{7} (40 x^{7}) d x

40 (\frac{78125 x ^{64}}{64} - \frac{171875 x ^{56}}{8} + \frac{2646875 x ^{48}}{16} - \frac{5823125 x ^{40}}{8} + \frac{64054375 x ^{32}}{32} - \frac{28183925 x ^{24}}{8} + \frac{62004635 x ^{16}}{16} - \frac{19487171 x ^{8}}{8}) + C

求解步骤

\int (5 x^{8} - 11)^{7} \cdot 40 x^{7} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 40 \cdot \int (5 x^{8} - 11)^{7} x^{7} d x

乘开

(5 x^{8} - 11)^{7} x^{7} : 78125 x^{63} - 1203125 x^{55} + 7940625 x^{47} - 29115625 x^{39} + 64054375 x^{31} - 84551775 x^{23} + 62004635 x^{15} - 19487171 x^{7}

= 40 \cdot \int 78125 x^{63} - 1203125 x^{55} + 7940625 x^{47} - 29115625 x^{39} + 64054375 x^{31} - 84551775 x^{23} + 62004635 x^{15} - 19487171 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

= 40 (\int 78125 x^{63} d x - \int 1203125 x^{55} d x + \int 7940625 x^{47} d x - \int 29115625 x^{39} d x + \int 64054375 x^{31} d x - \int 84551775 x^{23} d x + \int 62004635 x^{15} d x - \int 19487171 x^{7} d x)

\int 78125 x^{63} d x = \frac{78125 x ^{64}}{64}

\int 1203125 x^{55} d x = \frac{171875 x ^{56}}{8}

\int 7940625 x^{47} d x = \frac{2646875 x ^{48}}{16}

\int 29115625 x^{39} d x = \frac{5823125 x ^{40}}{8}

\int 64054375 x^{31} d x = \frac{64054375 x ^{32}}{32}

\int 84551775 x^{23} d x = \frac{28183925 x ^{24}}{8}

\int 62004635 x^{15} d x = \frac{62004635 x ^{16}}{16}

\int 19487171 x^{7} d x = \frac{19487171 x ^{8}}{8}

= 40 (\frac{78125 x ^{64}}{64} - \frac{171875 x ^{56}}{8} + \frac{2646875 x ^{48}}{16} - \frac{5823125 x ^{40}}{8} + \frac{64054375 x ^{32}}{32} - \frac{28183925 x ^{24}}{8} + \frac{62004635 x ^{16}}{16} - \frac{19487171 x ^{8}}{8})

解答补常数

= 40 (\frac{78125 x ^{64}}{64} - \frac{171875 x ^{56}}{8} + \frac{2646875 x ^{48}}{16} - \frac{5823125 x ^{40}}{8} + \frac{64054375 x ^{32}}{32} - \frac{28183925 x ^{24}}{8} + \frac{62004635 x ^{16}}{16} - \frac{19487171 x ^{8}}{8}) + C

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