integral of 5x^2(4x^3+7)^8

解答

\int 5 x^{2} (4 x^{3} + 7)^{8} d x

5 (\frac{65536 x ^{27}}{27} + \frac{114688 x ^{24}}{3} + \frac{802816 x ^{21}}{3} + \frac{9834496 x ^{18}}{9} + \frac{8605184 x ^{15}}{3} + \frac{15059072 x ^{12}}{3} + \frac{52706752 x ^{9}}{9} + \frac{13176688 x ^{6}}{3} + \frac{5764801 x ^{3}}{3}) + C

求解步骤

\int 5 x^{2} (4 x^{3} + 7)^{8} d x

提出常数:

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

= 5 \cdot \int x^{2} (4 x^{3} + 7)^{8} d x

乘开

x^{2} (4 x^{3} + 7)^{8} : 65536 x^{26} + 917504 x^{23} + 5619712 x^{20} + 19668992 x^{17} + 43025920 x^{14} + 60236288 x^{11} + 52706752 x^{8} + 26353376 x^{5} + 5764801 x^{2}

= 5 \cdot \int 65536 x^{26} + 917504 x^{23} + 5619712 x^{20} + 19668992 x^{17} + 43025920 x^{14} + 60236288 x^{11} + 52706752 x^{8} + 26353376 x^{5} + 5764801 x^{2} 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

= 5 (\int 65536 x^{26} d x + \int 917504 x^{23} d x + \int 5619712 x^{20} d x + \int 19668992 x^{17} d x + \int 43025920 x^{14} d x + \int 60236288 x^{11} d x + \int 52706752 x^{8} d x + \int 26353376 x^{5} d x + \int 5764801 x^{2} d x)

\int 65536 x^{26} d x = \frac{65536 x ^{27}}{27}

\int 917504 x^{23} d x = \frac{114688 x ^{24}}{3}

\int 5619712 x^{20} d x = \frac{802816 x ^{21}}{3}

\int 19668992 x^{17} d x = \frac{9834496 x ^{18}}{9}

\int 43025920 x^{14} d x = \frac{8605184 x ^{15}}{3}

\int 60236288 x^{11} d x = \frac{15059072 x ^{12}}{3}

\int 52706752 x^{8} d x = \frac{52706752 x ^{9}}{9}

\int 26353376 x^{5} d x = \frac{13176688 x ^{6}}{3}

\int 5764801 x^{2} d x = \frac{5764801 x ^{3}}{3}

= 5 (\frac{65536 x ^{27}}{27} + \frac{114688 x ^{24}}{3} + \frac{802816 x ^{21}}{3} + \frac{9834496 x ^{18}}{9} + \frac{8605184 x ^{15}}{3} + \frac{15059072 x ^{12}}{3} + \frac{52706752 x ^{9}}{9} + \frac{13176688 x ^{6}}{3} + \frac{5764801 x ^{3}}{3})

解答补常数

= 5 (\frac{65536 x ^{27}}{27} + \frac{114688 x ^{24}}{3} + \frac{802816 x ^{21}}{3} + \frac{9834496 x ^{18}}{9} + \frac{8605184 x ^{15}}{3} + \frac{15059072 x ^{12}}{3} + \frac{52706752 x ^{9}}{9} + \frac{13176688 x ^{6}}{3} + \frac{5764801 x ^{3}}{3}) + C

\int (4 x^{2} - 2 x)^{2} d x

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