integral from 0 to pi/2 of sin^2(z)

解

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

\frac{π}{4}

十進法表記

0.78539 \dots

解答ステップ

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

三角関数の公式を使用して書き換える

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

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{π}{2}} 1 - cos (2 z) d z

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{π}{2}} 1 d z - \int_{0}^{\frac{π}{2}} cos (2 z) d z)

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

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

= \frac{1}{2} (\frac{π}{2} - 0)

簡素化

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

= \frac{π}{4}