integral of 3x^2(x^3+4)^{10}

解答

\int 3 x^{2} (x^{3} + 4)^{10} d x

\frac{x ^{33}}{11} + 4 x^{30} + 80 x^{27} + 960 x^{24} + 7680 x^{21} + 43008 x^{18} + 172032 x^{15} + 491520 x^{12} + 983040 x^{9} + 1310720 x^{6} + 1048576 x^{3} + C

求解步骤

\int 3 x^{2} (x^{3} + 4)^{10} 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} (x^{3} + 4)^{10} d x

乘开

x^{2} (x^{3} + 4)^{10} : x^{32} + 40 x^{29} + 720 x^{26} + 7680 x^{23} + 53760 x^{20} + 258048 x^{17} + 860160 x^{14} + 1966080 x^{11} + 2949120 x^{8} + 2621440 x^{5} + 1048576 x^{2}

= 3 \cdot \int x^{32} + 40 x^{29} + 720 x^{26} + 7680 x^{23} + 53760 x^{20} + 258048 x^{17} + 860160 x^{14} + 1966080 x^{11} + 2949120 x^{8} + 2621440 x^{5} + 1048576 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

= 3 (\int x^{32} d x + \int 40 x^{29} d x + \int 720 x^{26} d x + \int 7680 x^{23} d x + \int 53760 x^{20} d x + \int 258048 x^{17} d x + \int 860160 x^{14} d x + \int 1966080 x^{11} d x + \int 2949120 x^{8} d x + \int 2621440 x^{5} d x + \int 1048576 x^{2} d x)

\int x^{32} d x = \frac{x ^{33}}{33}

\int 40 x^{29} d x = \frac{4 x ^{30}}{3}

\int 720 x^{26} d x = \frac{80 x ^{27}}{3}

\int 7680 x^{23} d x = 320 x^{24}

\int 53760 x^{20} d x = 2560 x^{21}

\int 258048 x^{17} d x = 14336 x^{18}

\int 860160 x^{14} d x = 57344 x^{15}

\int 1966080 x^{11} d x = 163840 x^{12}

\int 2949120 x^{8} d x = 327680 x^{9}

\int 2621440 x^{5} d x = \frac{1310720 x ^{6}}{3}

\int 1048576 x^{2} d x = \frac{1048576 x ^{3}}{3}

= 3 (\frac{x ^{33}}{33} + \frac{4 x ^{30}}{3} + \frac{80 x ^{27}}{3} + 320 x^{24} + 2560 x^{21} + 14336 x^{18} + 57344 x^{15} + 163840 x^{12} + 327680 x^{9} + \frac{1310720 x ^{6}}{3} + \frac{1048576 x ^{3}}{3})

化简

3 (\frac{x ^{33}}{33} + \frac{4 x ^{30}}{3} + \frac{80 x ^{27}}{3} + 320 x^{24} + 2560 x^{21} + 14336 x^{18} + 57344 x^{15} + 163840 x^{12} + 327680 x^{9} + \frac{1310720 x ^{6}}{3} + \frac{1048576 x ^{3}}{3}) : \frac{x ^{33}}{11} + 4 x^{30} + 80 x^{27} + 960 x^{24} + 7680 x^{21} + 43008 x^{18} + 172032 x^{15} + 491520 x^{12} + 983040 x^{9} + 1310720 x^{6} + 1048576 x^{3}

= \frac{x ^{33}}{11} + 4 x^{30} + 80 x^{27} + 960 x^{24} + 7680 x^{21} + 43008 x^{18} + 172032 x^{15} + 491520 x^{12} + 983040 x^{9} + 1310720 x^{6} + 1048576 x^{3}

解答补常数

= \frac{x ^{33}}{11} + 4 x^{30} + 80 x^{27} + 960 x^{24} + 7680 x^{21} + 43008 x^{18} + 172032 x^{15} + 491520 x^{12} + 983040 x^{9} + 1310720 x^{6} + 1048576 x^{3} + C

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