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人気のある微分積分 >

integral from 0 to pi/2 of (cos(x))^5

解

\int_{0}^{\frac{π}{2}} (cos (x))^{5} d x

解

\frac{8}{15}

+1

十進法表記

0.53333 \dots

解答ステップ

\int_{0}^{\frac{π}{2}} (cos (x))^{5} d x

簡素化

= \int_{0}^{\frac{π}{2}} cos^{5} (x) d x

三角関数の公式を使用して書き換える

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

u置換積分法を適用する

= \int_{0}^{1} (1 - u^{2})^{2} d u

拡張

(1 - u^{2})^{2} : 1 - 2 u^{2} + u^{4}

= \int_{0}^{1} 1 - 2 u^{2} + u^{4} 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

= \int_{0}^{1} 1 d u - \int_{0}^{1} 2 u^{2} d u + \int_{0}^{1} u^{4} d u

\int_{0}^{1} 1 d u = 1

\int_{0}^{1} 2 u^{2} d u = \frac{2}{3}

\int_{0}^{1} u^{4} d u = \frac{1}{5}

= 1 - \frac{2}{3} + \frac{1}{5}

簡素化

1 - \frac{2}{3} + \frac{1}{5} : \frac{8}{15}

= \frac{8}{15}

グラフ

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