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人気のある微分積分 >

integral from 0 to pi/2 of 11sin^2(2x)

解

\int_{0}^{\frac{π}{2}} 11 sin^{2} (2 x) d x

解

\frac{11 π}{4}

+1

十進法表記

8.63937 \dots

解答ステップ

\int_{0}^{\frac{π}{2}} 11 sin^{2} (2 x) d x

定数を除く:

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

= 11 \cdot \int_{0}^{\frac{π}{2}} sin^{2} (2 x) d x

三角関数の公式を使用して書き換える

= 11 \cdot \int_{0}^{\frac{π}{2}} \frac{1}{2} (1 - cos (4 x)) d x

定数を除く:

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

= 11 \cdot \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} 1 - cos (4 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

= 11 \cdot \frac{1}{2} (\int_{0}^{\frac{π}{2}} 1 d x - \int_{0}^{\frac{π}{2}} cos (4 x) d x)

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

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

= 11 \cdot \frac{1}{2} (\frac{π}{2} - 0)

簡素化

11 \cdot \frac{1}{2} (\frac{π}{2} - 0) : \frac{11 π}{4}

= \frac{11 π}{4}

グラフ

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