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人気のある微分積分 >

integral from 0 to 1/2 of (1-x^2)^{1/2}

解

\int_{0}^{\frac{1}{2}} (1 - x^{2})^{\frac{1}{2}} d x

解

\frac{2 π + 3 3}{24}

+1

十進法表記

0.47830 \dots

解答ステップ

\int_{0}^{\frac{1}{2}} (1 - x^{2})^{\frac{1}{2}} d x

三角関数による置換を適用する

= \int_{0}^{\frac{π}{6}} cos^{2} (u) d u

三角関数の公式を使用して書き換える

= \int_{0}^{\frac{π}{6}} \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{1}{2} \cdot \int_{0}^{\frac{π}{6}} 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{1}{2} (\int_{0}^{\frac{π}{6}} 1 d u + \int_{0}^{\frac{π}{6}} cos (2 u) d u)

\int_{0}^{\frac{π}{6}} 1 d u = \frac{π}{6}

\int_{0}^{\frac{π}{6}} cos (2 u) d u = \frac{3}{4}

= \frac{1}{2} (\frac{π}{6} + \frac{3}{4})

簡素化

\frac{1}{2} (\frac{π}{6} + \frac{3}{4}) : \frac{2 π + 3 3}{24}

= \frac{2 π + 3 3}{24}

グラフ

人気の例

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\int_{- 2 π}^{3 π} cos^{2} (x) d x

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\int_{1}^{3} \frac{1}{x ^{2} - 4 x + 5} d x

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\int_{- 1}^{3} (7 - 2 x) d x