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

解答

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

\frac{2 x ^{5}}{5} + \frac{3 x ^{4}}{4} + 2 x^{3} + \frac{9 x ^{2}}{2} + C

求解步骤

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

乘开

(x^{3} + 3 x) (2 x + 3) : 2 x^{4} + 3 x^{3} + 6 x^{2} + 9 x

= \int 2 x^{4} + 3 x^{3} + 6 x^{2} + 9 x 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^{4} d x + \int 3 x^{3} d x + \int 6 x^{2} d x + \int 9 x d x

\int 2 x^{4} d x = \frac{2 x ^{5}}{5}

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

\int 6 x^{2} d x = 2 x^{3}

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

= \frac{2 x ^{5}}{5} + \frac{3 x ^{4}}{4} + 2 x^{3} + \frac{9 x ^{2}}{2}

解答补常数

= \frac{2 x ^{5}}{5} + \frac{3 x ^{4}}{4} + 2 x^{3} + \frac{9 x ^{2}}{2} + C

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