integral of (2-sin(x/2))^2

解

\int (2 - sin (\frac{x}{2}))^{2} d x

\frac{3}{2} x + \frac{1}{4} sin (2 x) + 2 sin (x) + C

解答ステップ

\int (2 - sin (\frac{x}{2}))^{2} d x

三角関数の公式を使用して書き換える

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

拡張

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

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

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

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

\int 1 d x = x

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

簡素化

\frac{1}{2} (x + \frac{1}{2} sin (2 x)) + 2 sin (x) + x : \frac{3}{2} x + \frac{1}{4} sin (2 x) + 2 sin (x)

= \frac{3}{2} x + \frac{1}{4} sin (2 x) + 2 sin (x)

定数を解答に追加する

= \frac{3}{2} x + \frac{1}{4} sin (2 x) + 2 sin (x) + C