积分 (5-3x)(2+x^2)

解答

积分

(5 - 3 x) (2 + x^{2})

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

求解步骤

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

乘开

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

= \int 10 + 5 x^{2} - 6 x - 3 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 10 d x + \int 5 x^{2} d x - \int 6 x d x - \int 3 x^{3} d x

\int 10 d x = 10 x

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

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

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

= 10 x + \frac{5 x ^{3}}{3} - 3 x^{2} - \frac{3 x ^{4}}{4}

解答补常数

= 10 x + \frac{5 x ^{3}}{3} - 3 x^{2} - \frac{3 x ^{4}}{4} + C

(d^{4} - 2 d^{2} + 1) y = cos (x)

(d^{4} - 1) y = sin (x)

(d^{2} - 3 d + 2) y = x \cdot e^{x}

y^{^{''}} - 4 y^{^{'}} + 4 y = 12 e^{2 x} + x^{2}

(d^{3} - 2 d + 4) y = e^{x} cos (x)