integral of (5u-1)(3u^3+2)

解答

\int (5 u - 1) (3 u^{3} + 2) d u

3 u^{5} + 5 u^{2} - \frac{3 u ^{4}}{4} - 2 u + C

求解步骤

\int (5 u - 1) (3 u^{3} + 2) d u

乘开

(5 u - 1) (3 u^{3} + 2) : 15 u^{4} + 10 u - 3 u^{3} - 2

= \int 15 u^{4} + 10 u - 3 u^{3} - 2 d u

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 15 u^{4} d u + \int 10 u d u - \int 3 u^{3} d u - \int 2 d u

\int 15 u^{4} d u = 3 u^{5}

\int 10 u d u = 5 u^{2}

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

\int 2 d u = 2 u

= 3 u^{5} + 5 u^{2} - \frac{3 u ^{4}}{4} - 2 u

解答补常数

= 3 u^{5} + 5 u^{2} - \frac{3 u ^{4}}{4} - 2 u + C

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