integral from 0 to a/2 of sin^2((pix)/a)

解

\int_{0}^{\frac{a}{2}} sin^{2} (\frac{π x}{a}) d x

\frac{a}{4}

解答ステップ

\int_{0}^{\frac{a}{2}} sin^{2} (\frac{π x}{a}) d x

u置換積分法を適用する

= \int_{0}^{\frac{π}{2}} \frac{a sin ^{2} ( u )}{π} d u

定数を除く:

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

= \frac{a}{π} \cdot \int_{0}^{\frac{π}{2}} sin^{2} (u) d u

三角関数の公式を使用して書き換える

= \frac{a}{π} \cdot \int_{0}^{\frac{π}{2}} \frac{1 - cos ( 2 u )}{2} d u

定数を除く:

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

= \frac{a}{π} \cdot \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} 1 - cos (2 u) d u

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{a}{π} \cdot \frac{1}{2} (\int_{0}^{\frac{π}{2}} 1 d u - \int_{0}^{\frac{π}{2}} cos (2 u) d u)

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

\int_{0}^{\frac{π}{2}} cos (2 u) d u = 0

= \frac{a}{π} \cdot \frac{1}{2} (\frac{π}{2} - 0)

簡素化

\frac{a}{π} \cdot \frac{1}{2} (\frac{π}{2} - 0) : \frac{a}{4}

= \frac{a}{4}