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人気のある微分積分 >

integral from 0 to 1/4 of 4arcsin(4y)

解

\int_{0}^{\frac{1}{4}} 4 arcsin (4 y) d y

解

\frac{π}{2} - 1

+1

十進法表記

0.57079 \dots

解答ステップ

\int_{0}^{\frac{1}{4}} 4 arcsin (4 y) d y

定数を除く:

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

= 4 \cdot \int_{0}^{\frac{1}{4}} arcsin (4 y) d y

部分積分を適用する

= 4 [y arcsin (4 y) - \int \frac{4 y}{1 - 16 y ^{2}} d y]_{0}^{\frac{1}{4}}

\int \frac{4 y}{1 - 16 y ^{2}} d y = - \frac{1}{4} 1 - 16 y^{2}

= 4 [y arcsin (4 y) - (- \frac{1}{4} 1 - 16 y^{2})]_{0}^{\frac{1}{4}}

簡素化

4 [y arcsin (4 y) - (- \frac{1}{4} 1 - 16 y^{2})]_{0}^{\frac{1}{4}} : 4 [\frac{1}{4} (4 y arcsin (4 y) + 1 - 16 y^{2})]_{0}^{\frac{1}{4}}

= 4 [\frac{1}{4} (4 y arcsin (4 y) + 1 - 16 y^{2})]_{0}^{\frac{1}{4}}

限度を計算する:

\frac{π}{8} - \frac{1}{4}

= 4 (\frac{π}{8} - \frac{1}{4})

簡素化

= \frac{π}{2} - 1

グラフ

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