integral of (x^2+3x+1)^3(2x+3)

解答

\int (x^{2} + 3 x + 1)^{3} (2 x + 3) d x

\frac{x ^{8}}{4} + 3 x^{7} + \frac{29 x ^{6}}{2} + 36 x^{5} + \frac{195 x ^{4}}{4} + 36 x^{3} + \frac{29 x ^{2}}{2} + 3 x + C

求解步骤

\int (x^{2} + 3 x + 1)^{3} (2 x + 3) d x

乘开

(x^{2} + 3 x + 1)^{3} (2 x + 3) : 2 x^{7} + 21 x^{6} + 87 x^{5} + 180 x^{4} + 195 x^{3} + 108 x^{2} + 29 x + 3

= \int 2 x^{7} + 21 x^{6} + 87 x^{5} + 180 x^{4} + 195 x^{3} + 108 x^{2} + 29 x + 3 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^{7} d x + \int 21 x^{6} d x + \int 87 x^{5} d x + \int 180 x^{4} d x + \int 195 x^{3} d x + \int 108 x^{2} d x + \int 29 x d x + \int 3 d x

\int 2 x^{7} d x = \frac{x ^{8}}{4}

\int 21 x^{6} d x = 3 x^{7}

\int 87 x^{5} d x = \frac{29 x ^{6}}{2}

\int 180 x^{4} d x = 36 x^{5}

\int 195 x^{3} d x = \frac{195 x ^{4}}{4}

\int 108 x^{2} d x = 36 x^{3}

\int 29 x d x = \frac{29 x ^{2}}{2}

\int 3 d x = 3 x

= \frac{x ^{8}}{4} + 3 x^{7} + \frac{29 x ^{6}}{2} + 36 x^{5} + \frac{195 x ^{4}}{4} + 36 x^{3} + \frac{29 x ^{2}}{2} + 3 x

解答补常数

= \frac{x ^{8}}{4} + 3 x^{7} + \frac{29 x ^{6}}{2} + 36 x^{5} + \frac{195 x ^{4}}{4} + 36 x^{3} + \frac{29 x ^{2}}{2} + 3 x + C

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