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

解

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

11 x + 12 sin (x) + sin (2 x) + C

解答ステップ

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

拡張

(3 + 2 cos (x))^{2} : 9 + 12 cos (x) + 4 cos^{2} (x)

= \int 9 + 12 cos (x) + 4 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 9 d x + \int 12 cos (x) d x + \int 4 cos^{2} (x) d x

\int 9 d x = 9 x

\int 12 cos (x) d x = 12 sin (x)

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

= 9 x + 12 sin (x) + 2 x + sin (2 x)

簡素化

= 11 x + 12 sin (x) + sin (2 x)

定数を解答に追加する

= 11 x + 12 sin (x) + sin (2 x) + C