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

解

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

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

解答ステップ

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

簡素化

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

三角関数の公式を使用して書き換える

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

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int 1 + cos (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

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

\int 1 d x = x

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

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

定数を解答に追加する

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