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

解答

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

3 x^{4} - \frac{56 x ^{3}}{3} + \frac{55 x ^{2}}{2} + 25 x + C

求解步骤

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

乘开

(3 x + 1) (2 x - 5)^{2} : 12 x^{3} - 56 x^{2} + 55 x + 25

= \int 12 x^{3} - 56 x^{2} + 55 x + 25 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 12 x^{3} d x - \int 56 x^{2} d x + \int 55 x d x + \int 25 d x

\int 12 x^{3} d x = 3 x^{4}

\int 56 x^{2} d x = \frac{56 x ^{3}}{3}

\int 55 x d x = \frac{55 x ^{2}}{2}

\int 25 d x = 25 x

= 3 x^{4} - \frac{56 x ^{3}}{3} + \frac{55 x ^{2}}{2} + 25 x

解答补常数

= 3 x^{4} - \frac{56 x ^{3}}{3} + \frac{55 x ^{2}}{2} + 25 x + C

\int \frac{1}{( x - 1 ) ( x + 2 )} d x

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\int arctan (a x) d x