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

解答

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

729 x^{36} + 2187 x^{35} + 2916 x^{34} + 2268 x^{33} + 1134 x^{32} + 378 x^{31} + 84 x^{30} + 12 x^{29} + x^{28} + \frac{x ^{27}}{27} + C

求解步骤

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

乘开

(4 x^{3} + x^{2}) (3 x^{4} + x^{3})^{8} : 26244 x^{35} + 76545 x^{34} + 99144 x^{33} + 74844 x^{32} + 36288 x^{31} + 11718 x^{30} + 2520 x^{29} + 348 x^{28} + 28 x^{27} + x^{26}

= \int 26244 x^{35} + 76545 x^{34} + 99144 x^{33} + 74844 x^{32} + 36288 x^{31} + 11718 x^{30} + 2520 x^{29} + 348 x^{28} + 28 x^{27} + 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

= \int 26244 x^{35} d x + \int 76545 x^{34} d x + \int 99144 x^{33} d x + \int 74844 x^{32} d x + \int 36288 x^{31} d x + \int 11718 x^{30} d x + \int 2520 x^{29} d x + \int 348 x^{28} d x + \int 28 x^{27} d x + \int x^{26} d x

\int 26244 x^{35} d x = 729 x^{36}

\int 76545 x^{34} d x = 2187 x^{35}

\int 99144 x^{33} d x = 2916 x^{34}

\int 74844 x^{32} d x = 2268 x^{33}

\int 36288 x^{31} d x = 1134 x^{32}

\int 11718 x^{30} d x = 378 x^{31}

\int 2520 x^{29} d x = 84 x^{30}

\int 348 x^{28} d x = 12 x^{29}

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

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

= 729 x^{36} + 2187 x^{35} + 2916 x^{34} + 2268 x^{33} + 1134 x^{32} + 378 x^{31} + 84 x^{30} + 12 x^{29} + x^{28} + \frac{x ^{27}}{27}

解答补常数

= 729 x^{36} + 2187 x^{35} + 2916 x^{34} + 2268 x^{33} + 1134 x^{32} + 378 x^{31} + 84 x^{30} + 12 x^{29} + x^{28} + \frac{x ^{27}}{27} + C

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