integral of (x^3+2x)sin(x)

解答

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

- x^{3} cos (x) + 3 x^{2} sin (x) + 4 x cos (x) - 4 sin (x) + C

求解步骤

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

乘开

(x^{3} + 2 x) sin (x) : x^{3} sin (x) + 2 x sin (x)

= \int x^{3} sin (x) + 2 x 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^{3} sin (x) d x + \int 2 x sin (x) d x

\int x^{3} sin (x) d x = - x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x)))

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

= - x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x))) + 2 (- x cos (x) + sin (x))

化简

- x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x))) + 2 (- x cos (x) + sin (x)) : - x^{3} cos (x) + 3 x^{2} sin (x) + 4 x cos (x) - 4 sin (x)

= - x^{3} cos (x) + 3 x^{2} sin (x) + 4 x cos (x) - 4 sin (x)

解答补常数

= - x^{3} cos (x) + 3 x^{2} sin (x) + 4 x cos (x) - 4 sin (x) + C

x \to 3 lim (7 + 6 (- 7))

\int 2 x + 9 d x

\frac{\partial}{\partial z} (x y)

\int_{0}^{\infty} e^{- x y} d y

\int 2 z^{5} e^{1 - z} d z