integral of (x^2-2)cos(2x)

解

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

\frac{1}{2} x^{2} sin (2 x) + \frac{1}{2} x cos (2 x) - \frac{5}{4} sin (2 x) + C

解答ステップ

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

拡張

(x^{2} - 2) cos (2 x) : x^{2} cos (2 x) - 2 cos (2 x)

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

\int x^{2} cos (2 x) d x = \frac{1}{2} x^{2} sin (2 x) + \frac{1}{2} x cos (2 x) - \frac{1}{4} sin (2 x)

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

= \frac{1}{2} x^{2} sin (2 x) + \frac{1}{2} x cos (2 x) - \frac{1}{4} sin (2 x) - sin (2 x)

類似した元を足す:

- \frac{1}{4} sin (2 x) - sin (2 x) = - \frac{5}{4} sin (2 x)

= \frac{1}{2} x^{2} sin (2 x) + \frac{1}{2} x cos (2 x) - \frac{5}{4} sin (2 x)

定数を解答に追加する

= \frac{1}{2} x^{2} sin (2 x) + \frac{1}{2} x cos (2 x) - \frac{5}{4} sin (2 x) + C