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人気のある微分積分 >

integral from 0 to pi/4 of 4sin^2(x)

解

\int_{0}^{\frac{π}{4}} 4 sin^{2} (x) d x

解

\frac{π}{2} - 1

+1

十進法表記

0.57079 \dots

解答ステップ

\int_{0}^{\frac{π}{4}} 4 sin^{2} (x) d x

定数を除く:

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

= 4 \cdot \int_{0}^{\frac{π}{4}} sin^{2} (x) d x

三角関数の公式を使用して書き換える

= 4 \cdot \int_{0}^{\frac{π}{4}} \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

= 4 \cdot \frac{1}{2} \cdot \int_{0}^{\frac{π}{4}} 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

= 4 \cdot \frac{1}{2} (\int_{0}^{\frac{π}{4}} 1 d x - \int_{0}^{\frac{π}{4}} cos (2 x) d x)

\int_{0}^{\frac{π}{4}} 1 d x = \frac{π}{4}

\int_{0}^{\frac{π}{4}} cos (2 x) d x = \frac{1}{2}

= 4 \cdot \frac{1}{2} (\frac{π}{4} - \frac{1}{2})

簡素化

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

= \frac{π}{2} - 1

グラフ

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