integral of (x^3+x^2)^9(3x^2+2x)

解答

\int (x^{3} + x^{2})^{9} (3 x^{2} + 2 x) d x

\frac{x ^{30}}{10} + x^{29} + \frac{9 x ^{28}}{2} + 12 x^{27} + 21 x^{26} + \frac{126 x ^{25}}{5} + 21 x^{24} + 12 x^{23} + \frac{9 x ^{22}}{2} + x^{21} + \frac{x ^{20}}{10} + C

求解步骤

\int (x^{3} + x^{2})^{9} (3 x^{2} + 2 x) d x

乘开

(x^{3} + x^{2})^{9} (3 x^{2} + 2 x) : 3 x^{29} + 29 x^{28} + 126 x^{27} + 324 x^{26} + 546 x^{25} + 630 x^{24} + 504 x^{23} + 276 x^{22} + 99 x^{21} + 21 x^{20} + 2 x^{19}

= \int 3 x^{29} + 29 x^{28} + 126 x^{27} + 324 x^{26} + 546 x^{25} + 630 x^{24} + 504 x^{23} + 276 x^{22} + 99 x^{21} + 21 x^{20} + 2 x^{19} 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^{29} d x + \int 29 x^{28} d x + \int 126 x^{27} d x + \int 324 x^{26} d x + \int 546 x^{25} d x + \int 630 x^{24} d x + \int 504 x^{23} d x + \int 276 x^{22} d x + \int 99 x^{21} d x + \int 21 x^{20} d x + \int 2 x^{19} d x

\int 3 x^{29} d x = \frac{x ^{30}}{10}

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

\int 126 x^{27} d x = \frac{9 x ^{28}}{2}

\int 324 x^{26} d x = 12 x^{27}

\int 546 x^{25} d x = 21 x^{26}

\int 630 x^{24} d x = \frac{126 x ^{25}}{5}

\int 504 x^{23} d x = 21 x^{24}

\int 276 x^{22} d x = 12 x^{23}

\int 99 x^{21} d x = \frac{9 x ^{22}}{2}

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

\int 2 x^{19} d x = \frac{x ^{20}}{10}

= \frac{x ^{30}}{10} + x^{29} + \frac{9 x ^{28}}{2} + 12 x^{27} + 21 x^{26} + \frac{126 x ^{25}}{5} + 21 x^{24} + 12 x^{23} + \frac{9 x ^{22}}{2} + x^{21} + \frac{x ^{20}}{10}

解答补常数

= \frac{x ^{30}}{10} + x^{29} + \frac{9 x ^{28}}{2} + 12 x^{27} + 21 x^{26} + \frac{126 x ^{25}}{5} + 21 x^{24} + 12 x^{23} + \frac{9 x ^{22}}{2} + x^{21} + \frac{x ^{20}}{10} + C

\int 7 e^{4 x} cos (8 x) d x

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x \to \infty lim (5.3)