integral of ([(3x)]^2+2x)(x^3+x^2)

解答

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

\frac{3 x ^{6}}{2} + \frac{11 x ^{5}}{5} + \frac{x ^{4}}{2} + C

求解步骤

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

乘开

([3 x]^{2} + 2 x) (x^{3} + x^{2}) : 9 x^{5} + 11 x^{4} + 2 x^{3}

= \int 9 x^{5} + 11 x^{4} + 2 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 9 x^{5} d x + \int 11 x^{4} d x + \int 2 x^{3} d x

\int 9 x^{5} d x = \frac{3 x ^{6}}{2}

\int 11 x^{4} d x = \frac{11 x ^{5}}{5}

\int 2 x^{3} d x = \frac{x ^{4}}{2}

= \frac{3 x ^{6}}{2} + \frac{11 x ^{5}}{5} + \frac{x ^{4}}{2}

解答补常数

= \frac{3 x ^{6}}{2} + \frac{11 x ^{5}}{5} + \frac{x ^{4}}{2} + C

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