integral of (6x+4)cos(x/4)

解

\int (6 x + 4) cos (\frac{x}{4}) d x

6 (4 x sin (\frac{x}{4}) + 16 cos (\frac{x}{4})) + 16 sin (\frac{x}{4}) + C

解答ステップ

\int (6 x + 4) cos (\frac{x}{4}) d x

拡張

(6 x + 4) cos (\frac{x}{4}) : 6 x cos (\frac{x}{4}) + 4 cos (\frac{x}{4})

= \int 6 x cos (\frac{x}{4}) + 4 cos (\frac{x}{4}) 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 6 x cos (\frac{x}{4}) d x + \int 4 cos (\frac{x}{4}) d x

\int 6 x cos (\frac{x}{4}) d x = 6 (4 x sin (\frac{x}{4}) + 16 cos (\frac{x}{4}))

\int 4 cos (\frac{x}{4}) d x = 16 sin (\frac{x}{4})

= 6 (4 x sin (\frac{x}{4}) + 16 cos (\frac{x}{4})) + 16 sin (\frac{x}{4})

定数を解答に追加する

= 6 (4 x sin (\frac{x}{4}) + 16 cos (\frac{x}{4})) + 16 sin (\frac{x}{4}) + C