integral of (2x+4)(x^2+4x+10)^4

解答

\int (2 x + 4) (x^{2} + 4 x + 10)^{4} d x

\frac{x ^{10}}{5} + 4 x^{9} + 42 x^{8} + 288 x^{7} + 1416 x^{6} + \frac{25824 x ^{5}}{5} + 14160 x^{4} + 28800 x^{3} + 42000 x^{2} + 40000 x + C

求解步骤

\int (2 x + 4) (x^{2} + 4 x + 10)^{4} d x

乘开

(2 x + 4) (x^{2} + 4 x + 10)^{4} : 2 x^{9} + 36 x^{8} + 336 x^{7} + 2016 x^{6} + 8496 x^{5} + 25824 x^{4} + 56640 x^{3} + 86400 x^{2} + 84000 x + 40000

= \int 2 x^{9} + 36 x^{8} + 336 x^{7} + 2016 x^{6} + 8496 x^{5} + 25824 x^{4} + 56640 x^{3} + 86400 x^{2} + 84000 x + 40000 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 2 x^{9} d x + \int 36 x^{8} d x + \int 336 x^{7} d x + \int 2016 x^{6} d x + \int 8496 x^{5} d x + \int 25824 x^{4} d x + \int 56640 x^{3} d x + \int 86400 x^{2} d x + \int 84000 x d x + \int 40000 d x

\int 2 x^{9} d x = \frac{x ^{10}}{5}

\int 36 x^{8} d x = 4 x^{9}

\int 336 x^{7} d x = 42 x^{8}

\int 2016 x^{6} d x = 288 x^{7}

\int 8496 x^{5} d x = 1416 x^{6}

\int 25824 x^{4} d x = \frac{25824 x ^{5}}{5}

\int 56640 x^{3} d x = 14160 x^{4}

\int 86400 x^{2} d x = 28800 x^{3}

\int 84000 x d x = 42000 x^{2}

\int 40000 d x = 40000 x

= \frac{x ^{10}}{5} + 4 x^{9} + 42 x^{8} + 288 x^{7} + 1416 x^{6} + \frac{25824 x ^{5}}{5} + 14160 x^{4} + 28800 x^{3} + 42000 x^{2} + 40000 x

解答补常数

= \frac{x ^{10}}{5} + 4 x^{9} + 42 x^{8} + 288 x^{7} + 1416 x^{6} + \frac{25824 x ^{5}}{5} + 14160 x^{4} + 28800 x^{3} + 42000 x^{2} + 40000 x + C

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