integral from 0 to 1 of cos^2(ax)

解

\int_{0}^{1} cos^{2} (a x) d x

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

解答ステップ

\int_{0}^{1} cos^{2} (a x) d x

三角関数の公式を使用して書き換える

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

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

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

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