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

解答

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

\frac{x ^{36}}{9} + x^{34} + 4 x^{32} + \frac{28 x ^{30}}{3} + 14 x^{28} + 14 x^{26} + \frac{28 x ^{24}}{3} + 4 x^{22} + x^{20} + \frac{x ^{18}}{9} + C

求解步骤

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

乘开

(x^{4} + x^{2})^{8} (4 x^{3} + 2 x) : 4 x^{35} + 34 x^{33} + 128 x^{31} + 280 x^{29} + 392 x^{27} + 364 x^{25} + 224 x^{23} + 88 x^{21} + 20 x^{19} + 2 x^{17}

= \int 4 x^{35} + 34 x^{33} + 128 x^{31} + 280 x^{29} + 392 x^{27} + 364 x^{25} + 224 x^{23} + 88 x^{21} + 20 x^{19} + 2 x^{17} 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 4 x^{35} d x + \int 34 x^{33} d x + \int 128 x^{31} d x + \int 280 x^{29} d x + \int 392 x^{27} d x + \int 364 x^{25} d x + \int 224 x^{23} d x + \int 88 x^{21} d x + \int 20 x^{19} d x + \int 2 x^{17} d x

\int 4 x^{35} d x = \frac{x ^{36}}{9}

\int 34 x^{33} d x = x^{34}

\int 128 x^{31} d x = 4 x^{32}

\int 280 x^{29} d x = \frac{28 x ^{30}}{3}

\int 392 x^{27} d x = 14 x^{28}

\int 364 x^{25} d x = 14 x^{26}

\int 224 x^{23} d x = \frac{28 x ^{24}}{3}

\int 88 x^{21} d x = 4 x^{22}

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

\int 2 x^{17} d x = \frac{x ^{18}}{9}

= \frac{x ^{36}}{9} + x^{34} + 4 x^{32} + \frac{28 x ^{30}}{3} + 14 x^{28} + 14 x^{26} + \frac{28 x ^{24}}{3} + 4 x^{22} + x^{20} + \frac{x ^{18}}{9}

解答补常数

= \frac{x ^{36}}{9} + x^{34} + 4 x^{32} + \frac{28 x ^{30}}{3} + 14 x^{28} + 14 x^{26} + \frac{28 x ^{24}}{3} + 4 x^{22} + x^{20} + \frac{x ^{18}}{9} + C

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