integral from 0 to (2pi)/w of cos^2(wt)

解

\int_{0}^{\frac{2 π}{w}} cos^{2} (wt) d t

\frac{π}{w}

解答ステップ

\int_{0}^{\frac{2 π}{w}} cos^{2} (wt) d t

三角関数の公式を使用して書き換える

= \int_{0}^{\frac{2 π}{w}} \frac{1 + cos ( 2 wt )}{2} d t

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}^{\frac{2 π}{w}} 1 + cos (2 wt) d t

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}^{\frac{2 π}{w}} 1 d t + \int_{0}^{\frac{2 π}{w}} cos (2 wt) d t)

\int_{0}^{\frac{2 π}{w}} 1 d t = \frac{2 π}{w}

\int_{0}^{\frac{2 π}{w}} cos (2 wt) d t = 0

= \frac{1}{2} (\frac{2 π}{w} + 0)

簡素化

\frac{1}{2} (\frac{2 π}{w} + 0) : \frac{π}{w}

= \frac{π}{w}