integral of x^4(3x^5-1)^6

解答

\int x^{4} (3 x^{5} - 1)^{6} d x

\frac{729 x ^{35}}{35} - \frac{243 x ^{30}}{5} + \frac{243 x ^{25}}{5} - 27 x^{20} + 9 x^{15} - \frac{9 x ^{10}}{5} + \frac{x ^{5}}{5} + C

求解步骤

\int x^{4} (3 x^{5} - 1)^{6} d x

乘开

x^{4} (3 x^{5} - 1)^{6} : 729 x^{34} - 1458 x^{29} + 1215 x^{24} - 540 x^{19} + 135 x^{14} - 18 x^{9} + x^{4}

= \int 729 x^{34} - 1458 x^{29} + 1215 x^{24} - 540 x^{19} + 135 x^{14} - 18 x^{9} + x^{4} 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 729 x^{34} d x - \int 1458 x^{29} d x + \int 1215 x^{24} d x - \int 540 x^{19} d x + \int 135 x^{14} d x - \int 18 x^{9} d x + \int x^{4} d x

\int 729 x^{34} d x = \frac{729 x ^{35}}{35}

\int 1458 x^{29} d x = \frac{243 x ^{30}}{5}

\int 1215 x^{24} d x = \frac{243 x ^{25}}{5}

\int 540 x^{19} d x = 27 x^{20}

\int 135 x^{14} d x = 9 x^{15}

\int 18 x^{9} d x = \frac{9 x ^{10}}{5}

\int x^{4} d x = \frac{x ^{5}}{5}

= \frac{729 x ^{35}}{35} - \frac{243 x ^{30}}{5} + \frac{243 x ^{25}}{5} - 27 x^{20} + 9 x^{15} - \frac{9 x ^{10}}{5} + \frac{x ^{5}}{5}

解答补常数

= \frac{729 x ^{35}}{35} - \frac{243 x ^{30}}{5} + \frac{243 x ^{25}}{5} - 27 x^{20} + 9 x^{15} - \frac{9 x ^{10}}{5} + \frac{x ^{5}}{5} + C

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