integral of (x^3-3x+8)cos(x)

解

\int (x^{3} - 3 x + 8) cos (x) d x

x^{3} sin (x) + 3 x^{2} cos (x) - 9 x sin (x) - 9 cos (x) + 8 sin (x) + C

解答ステップ

\int (x^{3} - 3 x + 8) cos (x) d x

拡張

(x^{3} - 3 x + 8) cos (x) : x^{3} cos (x) - 3 x cos (x) + 8 cos (x)

= \int x^{3} cos (x) - 3 x cos (x) + 8 cos (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} cos (x) d x - \int 3 x cos (x) d x + \int 8 cos (x) d x

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

\int 3 x cos (x) d x = 3 (x sin (x) + cos (x))

\int 8 cos (x) d x = 8 sin (x)

= x^{3} sin (x) - 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x))) - 3 (x sin (x) + cos (x)) + 8 sin (x)

簡素化

x^{3} sin (x) - 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x))) - 3 (x sin (x) + cos (x)) + 8 sin (x) : x^{3} sin (x) + 3 x^{2} cos (x) - 9 x sin (x) - 9 cos (x) + 8 sin (x)

= x^{3} sin (x) + 3 x^{2} cos (x) - 9 x sin (x) - 9 cos (x) + 8 sin (x)

定数を解答に追加する

= x^{3} sin (x) + 3 x^{2} cos (x) - 9 x sin (x) - 9 cos (x) + 8 sin (x) + C