integral of (2x^3+2x)cos(x^2+1)

解

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

x^{2} sin (x^{2} + 1) + cos (x^{2} + 1) + sin (x^{2} + 1) + C

解答ステップ

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

拡張

(2 x^{3} + 2 x) cos (x^{2} + 1) : 2 x^{3} cos (x^{2} + 1) + 2 x cos (x^{2} + 1)

= \int 2 x^{3} cos (x^{2} + 1) + 2 x cos (x^{2} + 1) 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 2 x^{3} cos (x^{2} + 1) d x + \int 2 x cos (x^{2} + 1) d x

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

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

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

定数を解答に追加する

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