integral of (x^2+1)*sin(x)

解

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

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

解答ステップ

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

部分積分を適用する

= - cos (x) (x^{2} + 1) - \int - 2 x cos (x) d x

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

= - cos (x) (x^{2} + 1) - (- 2 (x sin (x) + cos (x)))

簡素化

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

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

定数を解答に追加する

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