integral of (x^2+1)^9

解答

\int (x^{2} + 1)^{9} d x

\frac{x ^{19}}{19} + \frac{9 x ^{17}}{17} + \frac{12 x ^{15}}{5} + \frac{84 x ^{13}}{13} + \frac{126 x ^{11}}{11} + 14 x^{9} + 12 x^{7} + \frac{36 x ^{5}}{5} + 3 x^{3} + x + C

求解步骤

\int (x^{2} + 1)^{9} d x

乘开

(x^{2} + 1)^{9} : x^{18} + 9 x^{16} + 36 x^{14} + 84 x^{12} + 126 x^{10} + 126 x^{8} + 84 x^{6} + 36 x^{4} + 9 x^{2} + 1

= \int x^{18} + 9 x^{16} + 36 x^{14} + 84 x^{12} + 126 x^{10} + 126 x^{8} + 84 x^{6} + 36 x^{4} + 9 x^{2} + 1 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 x^{18} d x + \int 9 x^{16} d x + \int 36 x^{14} d x + \int 84 x^{12} d x + \int 126 x^{10} d x + \int 126 x^{8} d x + \int 84 x^{6} d x + \int 36 x^{4} d x + \int 9 x^{2} d x + \int 1 d x

\int x^{18} d x = \frac{x ^{19}}{19}

\int 9 x^{16} d x = \frac{9 x ^{17}}{17}

\int 36 x^{14} d x = \frac{12 x ^{15}}{5}

\int 84 x^{12} d x = \frac{84 x ^{13}}{13}

\int 126 x^{10} d x = \frac{126 x ^{11}}{11}

\int 126 x^{8} d x = 14 x^{9}

\int 84 x^{6} d x = 12 x^{7}

\int 36 x^{4} d x = \frac{36 x ^{5}}{5}

\int 9 x^{2} d x = 3 x^{3}

\int 1 d x = x

= \frac{x ^{19}}{19} + \frac{9 x ^{17}}{17} + \frac{12 x ^{15}}{5} + \frac{84 x ^{13}}{13} + \frac{126 x ^{11}}{11} + 14 x^{9} + 12 x^{7} + \frac{36 x ^{5}}{5} + 3 x^{3} + x

解答补常数

= \frac{x ^{19}}{19} + \frac{9 x ^{17}}{17} + \frac{12 x ^{15}}{5} + \frac{84 x ^{13}}{13} + \frac{126 x ^{11}}{11} + 14 x^{9} + 12 x^{7} + \frac{36 x ^{5}}{5} + 3 x^{3} + x + C

\int x^{2} x^{2} - 1 d x

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