integral of (5x^4+2x-3)cos(2x)

解

\int (5 x^{4} + 2 x - 3) cos (2 x) d x

\frac{5}{2} x^{4} sin (2 x) + 5 x^{3} cos (2 x) - \frac{15}{2} x^{2} sin (2 x) - \frac{15}{2} x cos (2 x) + x sin (2 x) + \frac{1}{2} cos (2 x) + \frac{9}{4} sin (2 x) + C

解答ステップ

\int (5 x^{4} + 2 x - 3) cos (2 x) d x

拡張

(5 x^{4} + 2 x - 3) cos (2 x) : 5 x^{4} cos (2 x) + 2 x cos (2 x) - 3 cos (2 x)

= \int 5 x^{4} cos (2 x) + 2 x cos (2 x) - 3 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 5 x^{4} cos (2 x) d x + \int 2 x cos (2 x) d x - \int 3 cos (2 x) d x

\int 5 x^{4} cos (2 x) d x = 5 (\frac{1}{2} x^{4} sin (2 x) + x^{3} cos (2 x) - \frac{3}{2} x^{2} sin (2 x) - \frac{3}{2} x cos (2 x) + \frac{3}{4} sin (2 x))

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

\int 3 cos (2 x) d x = \frac{3}{2} sin (2 x)

= 5 (\frac{1}{2} x^{4} sin (2 x) + x^{3} cos (2 x) - \frac{3}{2} x^{2} sin (2 x) - \frac{3}{2} x cos (2 x) + \frac{3}{4} sin (2 x)) + x sin (2 x) + \frac{1}{2} cos (2 x) - \frac{3}{2} sin (2 x)

簡素化

5 (\frac{1}{2} x^{4} sin (2 x) + x^{3} cos (2 x) - \frac{3}{2} x^{2} sin (2 x) - \frac{3}{2} x cos (2 x) + \frac{3}{4} sin (2 x)) + x sin (2 x) + \frac{1}{2} cos (2 x) - \frac{3}{2} sin (2 x) : \frac{5}{2} x^{4} sin (2 x) + 5 x^{3} cos (2 x) - \frac{15}{2} x^{2} sin (2 x) - \frac{15}{2} x cos (2 x) + x sin (2 x) + \frac{1}{2} cos (2 x) + \frac{9}{4} sin (2 x)

= \frac{5}{2} x^{4} sin (2 x) + 5 x^{3} cos (2 x) - \frac{15}{2} x^{2} sin (2 x) - \frac{15}{2} x cos (2 x) + x sin (2 x) + \frac{1}{2} cos (2 x) + \frac{9}{4} sin (2 x)

定数を解答に追加する

= \frac{5}{2} x^{4} sin (2 x) + 5 x^{3} cos (2 x) - \frac{15}{2} x^{2} sin (2 x) - \frac{15}{2} x cos (2 x) + x sin (2 x) + \frac{1}{2} cos (2 x) + \frac{9}{4} sin (2 x) + C