integral of (4+x^6)^6x^5

解答

\int (4 + x^{6})^{6} x^{5} d x

\frac{2048 x ^{6}}{3} + 512 x^{12} + \frac{640 x ^{18}}{3} + \frac{160 x ^{24}}{3} + 8 x^{30} + \frac{2 x ^{36}}{3} + \frac{x ^{42}}{42} + C

求解步骤

\int (4 + x^{6})^{6} x^{5} d x

乘开

(4 + x^{6})^{6} x^{5} : 4096 x^{5} + 6144 x^{11} + 3840 x^{17} + 1280 x^{23} + 240 x^{29} + 24 x^{35} + x^{41}

= \int 4096 x^{5} + 6144 x^{11} + 3840 x^{17} + 1280 x^{23} + 240 x^{29} + 24 x^{35} + x^{41} 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 4096 x^{5} d x + \int 6144 x^{11} d x + \int 3840 x^{17} d x + \int 1280 x^{23} d x + \int 240 x^{29} d x + \int 24 x^{35} d x + \int x^{41} d x

\int 4096 x^{5} d x = \frac{2048 x ^{6}}{3}

\int 6144 x^{11} d x = 512 x^{12}

\int 3840 x^{17} d x = \frac{640 x ^{18}}{3}

\int 1280 x^{23} d x = \frac{160 x ^{24}}{3}

\int 240 x^{29} d x = 8 x^{30}

\int 24 x^{35} d x = \frac{2 x ^{36}}{3}

\int x^{41} d x = \frac{x ^{42}}{42}

= \frac{2048 x ^{6}}{3} + 512 x^{12} + \frac{640 x ^{18}}{3} + \frac{160 x ^{24}}{3} + 8 x^{30} + \frac{2 x ^{36}}{3} + \frac{x ^{42}}{42}

解答补常数

= \frac{2048 x ^{6}}{3} + 512 x^{12} + \frac{640 x ^{18}}{3} + \frac{160 x ^{24}}{3} + 8 x^{30} + \frac{2 x ^{36}}{3} + \frac{x ^{42}}{42} + C

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