integral of 3x^5(5x^6-2)^9

解答

\int 3 x^{5} (5 x^{6} - 2)^{9} d x

\frac{390625 x ^{60}}{4} - 390625 x^{54} + 703125 x^{48} - 750000 x^{42} + 525000 x^{36} - 252000 x^{30} + 84000 x^{24} - 19200 x^{18} + 2880 x^{12} - 256 x^{6} + C

求解步骤

\int 3 x^{5} (5 x^{6} - 2)^{9} d x

提出常数:

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

= 3 \cdot \int x^{5} (5 x^{6} - 2)^{9} d x

乘开

x^{5} (5 x^{6} - 2)^{9} : 1953125 x^{59} - 7031250 x^{53} + 11250000 x^{47} - 10500000 x^{41} + 6300000 x^{35} - 2520000 x^{29} + 672000 x^{23} - 115200 x^{17} + 11520 x^{11} - 512 x^{5}

= 3 \cdot \int 1953125 x^{59} - 7031250 x^{53} + 11250000 x^{47} - 10500000 x^{41} + 6300000 x^{35} - 2520000 x^{29} + 672000 x^{23} - 115200 x^{17} + 11520 x^{11} - 512 x^{5} 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

= 3 (\int 1953125 x^{59} d x - \int 7031250 x^{53} d x + \int 11250000 x^{47} d x - \int 10500000 x^{41} d x + \int 6300000 x^{35} d x - \int 2520000 x^{29} d x + \int 672000 x^{23} d x - \int 115200 x^{17} d x + \int 11520 x^{11} d x - \int 512 x^{5} d x)

\int 1953125 x^{59} d x = \frac{390625 x ^{60}}{12}

\int 7031250 x^{53} d x = \frac{390625 x ^{54}}{3}

\int 11250000 x^{47} d x = 234375 x^{48}

\int 10500000 x^{41} d x = 250000 x^{42}

\int 6300000 x^{35} d x = 175000 x^{36}

\int 2520000 x^{29} d x = 84000 x^{30}

\int 672000 x^{23} d x = 28000 x^{24}

\int 115200 x^{17} d x = 6400 x^{18}

\int 11520 x^{11} d x = 960 x^{12}

\int 512 x^{5} d x = \frac{256 x ^{6}}{3}

= 3 (\frac{390625 x ^{60}}{12} - \frac{390625 x ^{54}}{3} + 234375 x^{48} - 250000 x^{42} + 175000 x^{36} - 84000 x^{30} + 28000 x^{24} - 6400 x^{18} + 960 x^{12} - \frac{256 x ^{6}}{3})

化简

3 (\frac{390625 x ^{60}}{12} - \frac{390625 x ^{54}}{3} + 234375 x^{48} - 250000 x^{42} + 175000 x^{36} - 84000 x^{30} + 28000 x^{24} - 6400 x^{18} + 960 x^{12} - \frac{256 x ^{6}}{3}) : \frac{390625 x ^{60}}{4} - 390625 x^{54} + 703125 x^{48} - 750000 x^{42} + 525000 x^{36} - 252000 x^{30} + 84000 x^{24} - 19200 x^{18} + 2880 x^{12} - 256 x^{6}

= \frac{390625 x ^{60}}{4} - 390625 x^{54} + 703125 x^{48} - 750000 x^{42} + 525000 x^{36} - 252000 x^{30} + 84000 x^{24} - 19200 x^{18} + 2880 x^{12} - 256 x^{6}

解答补常数

= \frac{390625 x ^{60}}{4} - 390625 x^{54} + 703125 x^{48} - 750000 x^{42} + 525000 x^{36} - 252000 x^{30} + 84000 x^{24} - 19200 x^{18} + 2880 x^{12} - 256 x^{6} + C