integral of 3x^2(2-x^3)^8

解答

\int 3 x^{2} (2 - x^{3})^{8} d x

256 x^{3} - 512 x^{6} + \frac{1792 x ^{9}}{3} - 448 x^{12} + 224 x^{15} - \frac{224 x ^{18}}{3} + 16 x^{21} - 2 x^{24} + \frac{x ^{27}}{9} + C

求解步骤

\int 3 x^{2} (2 - x^{3})^{8} d x

提出常数:

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

= 3 \cdot \int x^{2} (2 - x^{3})^{8} d x

乘开

x^{2} (2 - x^{3})^{8} : 256 x^{2} - 1024 x^{5} + 1792 x^{8} - 1792 x^{11} + 1120 x^{14} - 448 x^{17} + 112 x^{20} - 16 x^{23} + x^{26}

= 3 \cdot \int 256 x^{2} - 1024 x^{5} + 1792 x^{8} - 1792 x^{11} + 1120 x^{14} - 448 x^{17} + 112 x^{20} - 16 x^{23} + x^{26} 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 256 x^{2} d x - \int 1024 x^{5} d x + \int 1792 x^{8} d x - \int 1792 x^{11} d x + \int 1120 x^{14} d x - \int 448 x^{17} d x + \int 112 x^{20} d x - \int 16 x^{23} d x + \int x^{26} d x)

\int 256 x^{2} d x = \frac{256 x ^{3}}{3}

\int 1024 x^{5} d x = \frac{512 x ^{6}}{3}

\int 1792 x^{8} d x = \frac{1792 x ^{9}}{9}

\int 1792 x^{11} d x = \frac{448 x ^{12}}{3}

\int 1120 x^{14} d x = \frac{224 x ^{15}}{3}

\int 448 x^{17} d x = \frac{224 x ^{18}}{9}

\int 112 x^{20} d x = \frac{16 x ^{21}}{3}

\int 16 x^{23} d x = \frac{2 x ^{24}}{3}

\int x^{26} d x = \frac{x ^{27}}{27}

= 3 (\frac{256 x ^{3}}{3} - \frac{512 x ^{6}}{3} + \frac{1792 x ^{9}}{9} - \frac{448 x ^{12}}{3} + \frac{224 x ^{15}}{3} - \frac{224 x ^{18}}{9} + \frac{16 x ^{21}}{3} - \frac{2 x ^{24}}{3} + \frac{x ^{27}}{27})

化简

3 (\frac{256 x ^{3}}{3} - \frac{512 x ^{6}}{3} + \frac{1792 x ^{9}}{9} - \frac{448 x ^{12}}{3} + \frac{224 x ^{15}}{3} - \frac{224 x ^{18}}{9} + \frac{16 x ^{21}}{3} - \frac{2 x ^{24}}{3} + \frac{x ^{27}}{27}) : 256 x^{3} - 512 x^{6} + \frac{1792 x ^{9}}{3} - 448 x^{12} + 224 x^{15} - \frac{224 x ^{18}}{3} + 16 x^{21} - 2 x^{24} + \frac{x ^{27}}{9}

= 256 x^{3} - 512 x^{6} + \frac{1792 x ^{9}}{3} - 448 x^{12} + 224 x^{15} - \frac{224 x ^{18}}{3} + 16 x^{21} - 2 x^{24} + \frac{x ^{27}}{9}

解答补常数

= 256 x^{3} - 512 x^{6} + \frac{1792 x ^{9}}{3} - 448 x^{12} + 224 x^{15} - \frac{224 x ^{18}}{3} + 16 x^{21} - 2 x^{24} + \frac{x ^{27}}{9} + C

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