integral of sin(x)(3x^2+x-1)

解

\int sin (x) (3 x^{2} + x - 1) d x

- 3 x^{2} cos (x) + 6 x sin (x) - x cos (x) + 7 cos (x) + sin (x) + C

解答ステップ

\int sin (x) (3 x^{2} + x - 1) d x

拡張

sin (x) (3 x^{2} + x - 1) : 3 x^{2} sin (x) + x sin (x) - sin (x)

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

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

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

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

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

簡素化

3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x))) - x cos (x) + sin (x) - (- cos (x)) : - 3 x^{2} cos (x) + 6 x sin (x) - x cos (x) + 7 cos (x) + sin (x)

= - 3 x^{2} cos (x) + 6 x sin (x) - x cos (x) + 7 cos (x) + sin (x)

定数を解答に追加する

= - 3 x^{2} cos (x) + 6 x sin (x) - x cos (x) + 7 cos (x) + sin (x) + C