integral of (x^8-25)^9x^7

解答

\int (x^{8} - 25)^{9} x^{7} d x

\frac{x ^{80}}{80} - \frac{25 x ^{72}}{8} + \frac{5625 x ^{64}}{16} - \frac{46875 x ^{56}}{2} + \frac{8203125 x ^{48}}{8} - \frac{123046875 x ^{40}}{4} + \frac{5126953125 x ^{32}}{8} - \frac{2 5 ^{7} \cdot 3 x ^{24}}{2} + \frac{2 5 ^{8} \cdot 9 x ^{16}}{16} - \frac{2 5 ^{9} x ^{8}}{8} + C

求解步骤

\int (x^{8} - 25)^{9} x^{7} d x

乘开

(x^{8} - 25)^{9} x^{7} : x^{79} - 225 x^{71} + 22500 x^{63} - 1312500 x^{55} + 49218750 x^{47} - 1230468750 x^{39} + 20507812500 x^{31} - 2 5^{7} \cdot 36 x^{23} + 2 5^{8} \cdot 9 x^{15} - 2 5^{9} x^{7}

= \int x^{79} - 225 x^{71} + 22500 x^{63} - 1312500 x^{55} + 49218750 x^{47} - 1230468750 x^{39} + 20507812500 x^{31} - 2 5^{7} \cdot 36 x^{23} + 2 5^{8} \cdot 9 x^{15} - 2 5^{9} 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^{79} d x - \int 225 x^{71} d x + \int 22500 x^{63} d x - \int 1312500 x^{55} d x + \int 49218750 x^{47} d x - \int 1230468750 x^{39} d x + \int 20507812500 x^{31} d x - \int 2 5^{7} \cdot 36 x^{23} d x + \int 2 5^{8} \cdot 9 x^{15} d x - \int 2 5^{9} x^{7} d x

\int x^{79} d x = \frac{x ^{80}}{80}

\int 225 x^{71} d x = \frac{25 x ^{72}}{8}

\int 22500 x^{63} d x = \frac{5625 x ^{64}}{16}

\int 1312500 x^{55} d x = \frac{46875 x ^{56}}{2}

\int 49218750 x^{47} d x = \frac{8203125 x ^{48}}{8}

\int 1230468750 x^{39} d x = \frac{123046875 x ^{40}}{4}

\int 20507812500 x^{31} d x = \frac{5126953125 x ^{32}}{8}

\int 2 5^{7} \cdot 36 x^{23} d x = \frac{2 5 ^{7} \cdot 3 x ^{24}}{2}

\int 2 5^{8} \cdot 9 x^{15} d x = \frac{2 5 ^{8} \cdot 9 x ^{16}}{16}

\int 2 5^{9} x^{7} d x = \frac{2 5 ^{9} x ^{8}}{8}

= \frac{x ^{80}}{80} - \frac{25 x ^{72}}{8} + \frac{5625 x ^{64}}{16} - \frac{46875 x ^{56}}{2} + \frac{8203125 x ^{48}}{8} - \frac{123046875 x ^{40}}{4} + \frac{5126953125 x ^{32}}{8} - \frac{2 5 ^{7} \cdot 3 x ^{24}}{2} + \frac{2 5 ^{8} \cdot 9 x ^{16}}{16} - \frac{2 5 ^{9} x ^{8}}{8}

解答补常数

= \frac{x ^{80}}{80} - \frac{25 x ^{72}}{8} + \frac{5625 x ^{64}}{16} - \frac{46875 x ^{56}}{2} + \frac{8203125 x ^{48}}{8} - \frac{123046875 x ^{40}}{4} + \frac{5126953125 x ^{32}}{8} - \frac{2 5 ^{7} \cdot 3 x ^{24}}{2} + \frac{2 5 ^{8} \cdot 9 x ^{16}}{16} - \frac{2 5 ^{9} x ^{8}}{8} + C

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