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

解答

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

\frac{5 x ^{4}}{2} + 7 x^{3} - 3 x^{2} - 9 x + C

求解步骤

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

乘开

(2 x^{2} + 3 x - 3) (5 x + 3) : 10 x^{3} + 21 x^{2} - 6 x - 9

= \int 10 x^{3} + 21 x^{2} - 6 x - 9 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 10 x^{3} d x + \int 21 x^{2} d x - \int 6 x d x - \int 9 d x

\int 10 x^{3} d x = \frac{5 x ^{4}}{2}

\int 21 x^{2} d x = 7 x^{3}

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

\int 9 d x = 9 x

= \frac{5 x ^{4}}{2} + 7 x^{3} - 3 x^{2} - 9 x

解答补常数

= \frac{5 x ^{4}}{2} + 7 x^{3} - 3 x^{2} - 9 x + C

d er i v a t i v e f (x) = \frac{x ^{4}}{8} - 8 x

x \to \infty lim (5 + \frac{1}{x})

x \to 3 lim (4 + 4 (- 4))

\int 7 x e^{- 2 x} d x

\frac{d}{d x} (cos ((x^{2} + 6)^{5}))