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

解

\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

解答ステップ

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

拡張

(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

総和規則を適用する:

\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))

簡素化

- 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)

定数を解答に追加する

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