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人気のある微分積分 >

integral from 0 to pi/5 of sin^4(5x)

解

\int_{0}^{\frac{π}{5}} sin^{4} (5 x) d x

解

\frac{3 π}{40}

+1

十進法表記

0.23561 \dots

解答ステップ

\int_{0}^{\frac{π}{5}} sin^{4} (5 x) d x

u置換積分法を適用する

= \int_{0}^{π} sin^{4} (u) \frac{1}{5} d u

定数を除く:

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

= \frac{1}{5} \cdot \int_{0}^{π} sin^{4} (u) d u

積分換算を適用する

= \frac{1}{5} (- [\frac{cos ( u ) sin ^{3} ( u )}{4}]_{0}^{π} + \frac{3}{4} \cdot \int_{0}^{π} sin^{2} (u) d u)

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

= \frac{1}{5} (- [\frac{cos ( u ) sin ^{3} ( u )}{4}]_{0}^{π} + \frac{3}{4} \cdot \frac{π}{2})

簡素化

\frac{1}{5} (- [\frac{cos ( u ) sin ^{3} ( u )}{4}]_{0}^{π} + \frac{3}{4} \cdot \frac{π}{2}) : \frac{1}{5} (- [\frac{1}{4} sin^{3} (u) cos (u)]_{0}^{π} + \frac{3 π}{8})

= \frac{1}{5} (- [\frac{1}{4} sin^{3} (u) cos (u)]_{0}^{π} + \frac{3 π}{8})

限度を計算する:

0

= \frac{1}{5} (- 0 + \frac{3 π}{8})

簡素化

= \frac{3 π}{40}

グラフ

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