integral of (x^3+x^2)^7(3x^2+2x)

解答

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

\frac{x ^{24}}{8} + x^{23} + \frac{7 x ^{22}}{2} + 7 x^{21} + \frac{35 x ^{20}}{4} + 7 x^{19} + \frac{7 x ^{18}}{2} + x^{17} + \frac{x ^{16}}{8} + C

求解步骤

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

乘开

(x^{3} + x^{2})^{7} (3 x^{2} + 2 x) : 3 x^{23} + 23 x^{22} + 77 x^{21} + 147 x^{20} + 175 x^{19} + 133 x^{18} + 63 x^{17} + 17 x^{16} + 2 x^{15}

= \int 3 x^{23} + 23 x^{22} + 77 x^{21} + 147 x^{20} + 175 x^{19} + 133 x^{18} + 63 x^{17} + 17 x^{16} + 2 x^{15} 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 3 x^{23} d x + \int 23 x^{22} d x + \int 77 x^{21} d x + \int 147 x^{20} d x + \int 175 x^{19} d x + \int 133 x^{18} d x + \int 63 x^{17} d x + \int 17 x^{16} d x + \int 2 x^{15} d x

\int 3 x^{23} d x = \frac{x ^{24}}{8}

\int 23 x^{22} d x = x^{23}

\int 77 x^{21} d x = \frac{7 x ^{22}}{2}

\int 147 x^{20} d x = 7 x^{21}

\int 175 x^{19} d x = \frac{35 x ^{20}}{4}

\int 133 x^{18} d x = 7 x^{19}

\int 63 x^{17} d x = \frac{7 x ^{18}}{2}

\int 17 x^{16} d x = x^{17}

\int 2 x^{15} d x = \frac{x ^{16}}{8}

= \frac{x ^{24}}{8} + x^{23} + \frac{7 x ^{22}}{2} + 7 x^{21} + \frac{35 x ^{20}}{4} + 7 x^{19} + \frac{7 x ^{18}}{2} + x^{17} + \frac{x ^{16}}{8}

解答补常数

= \frac{x ^{24}}{8} + x^{23} + \frac{7 x ^{22}}{2} + 7 x^{21} + \frac{35 x ^{20}}{4} + 7 x^{19} + \frac{7 x ^{18}}{2} + x^{17} + \frac{x ^{16}}{8} + C

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