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

解答

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

- x^{2} cos (x) + 2 x sin (x) - 4 x cos (x) - 2 cos (x) + 4 sin (x) + C

求解步骤

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

乘开

(x + 2)^{2} sin (x) : x^{2} sin (x) + 4 x sin (x) + 4 sin (x)

= \int x^{2} sin (x) + 4 x sin (x) + 4 sin (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 x^{2} sin (x) d x + \int 4 x sin (x) d x + \int 4 sin (x) d x

\int x^{2} sin (x) d x = - x^{2} cos (x) + 2 (x sin (x) + cos (x))

\int 4 x sin (x) d x = 4 (- x cos (x) + sin (x))

\int 4 sin (x) d x = - 4 cos (x)

= - x^{2} cos (x) + 2 (x sin (x) + cos (x)) + 4 (- x cos (x) + sin (x)) - 4 cos (x)

化简

- x^{2} cos (x) + 2 (x sin (x) + cos (x)) + 4 (- x cos (x) + sin (x)) - 4 cos (x) : - x^{2} cos (x) + 2 x sin (x) - 4 x cos (x) - 2 cos (x) + 4 sin (x)

= - x^{2} cos (x) + 2 x sin (x) - 4 x cos (x) - 2 cos (x) + 4 sin (x)

解答补常数

= - x^{2} cos (x) + 2 x sin (x) - 4 x cos (x) - 2 cos (x) + 4 sin (x) + C

\frac{d}{d x} (x^{3} - x)

\int \frac{1}{5} - \frac{2}{x} d x

\frac{d y}{d x} = x x - 18

l a pl a ce t r an s f or m - 3 t sin (2 t)

d er i v a t i v e f (x) = 4 x^{2} + 2 x - 3