integral from 0 to x of cos^2(x)

解

\int_{0}^{x} cos^{2} (x) d x

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

解答ステップ

\int_{0}^{x} cos^{2} (x) d x

三角関数の公式を使用して書き換える

= \int_{0}^{x} \frac{1 + cos ( 2 x )}{2} 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_{0}^{x} 1 + 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

= \frac{1}{2} (\int_{0}^{x} 1 d x + \int_{0}^{x} cos (2 x) d x)

\int_{0}^{x} 1 d x = x

\int_{0}^{x} cos (2 x) d x = \frac{1}{2} sin (2 x)

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