integral of (-5x^4+8x^3)(-2x^5+4x^4)^6

解答

\int (- 5 x^{4} + 8 x^{3}) (- 2 x^{5} + 4 x^{4})^{6} d x

- \frac{64 x ^{35}}{7} + 128 x^{34} - 768 x^{33} + 2560 x^{32} - 5120 x^{31} + 6144 x^{30} - 4096 x^{29} + \frac{8192 x ^{28}}{7} + C

求解步骤

\int (- 5 x^{4} + 8 x^{3}) (- 2 x^{5} + 4 x^{4})^{6} d x

乘开

(- 5 x^{4} + 8 x^{3}) (- 2 x^{5} + 4 x^{4})^{6} : - 320 x^{34} + 4352 x^{33} - 25344 x^{32} + 81920 x^{31} - 158720 x^{30} + 184320 x^{29} - 118784 x^{28} + 32768 x^{27}

= \int - 320 x^{34} + 4352 x^{33} - 25344 x^{32} + 81920 x^{31} - 158720 x^{30} + 184320 x^{29} - 118784 x^{28} + 32768 x^{27} 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 320 x^{34} d x + \int 4352 x^{33} d x - \int 25344 x^{32} d x + \int 81920 x^{31} d x - \int 158720 x^{30} d x + \int 184320 x^{29} d x - \int 118784 x^{28} d x + \int 32768 x^{27} d x

\int 320 x^{34} d x = \frac{64 x ^{35}}{7}

\int 4352 x^{33} d x = 128 x^{34}

\int 25344 x^{32} d x = 768 x^{33}

\int 81920 x^{31} d x = 2560 x^{32}

\int 158720 x^{30} d x = 5120 x^{31}

\int 184320 x^{29} d x = 6144 x^{30}

\int 118784 x^{28} d x = 4096 x^{29}

\int 32768 x^{27} d x = \frac{8192 x ^{28}}{7}

= - \frac{64 x ^{35}}{7} + 128 x^{34} - 768 x^{33} + 2560 x^{32} - 5120 x^{31} + 6144 x^{30} - 4096 x^{29} + \frac{8192 x ^{28}}{7}

解答补常数

= - \frac{64 x ^{35}}{7} + 128 x^{34} - 768 x^{33} + 2560 x^{32} - 5120 x^{31} + 6144 x^{30} - 4096 x^{29} + \frac{8192 x ^{28}}{7} + C

x \to 2 lim (\frac{( 3 x + 2 )}{5 x - 2})^{\frac{( 3 )}{x - 2}}

x \to \infty lim (x^{6} e^{- x^{5}})

\int sinh (1 + 4 x) d x

l a pl a ce t r an s f or m (t^{2} + \frac{1}{2})^{2}

(x^{2} + 2 y^{2}) d x - x y d y = 0