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人気のある微分積分 >

integral from 0 to pi/2 of 15cos^5(x)

解

\int_{0}^{\frac{π}{2}} 15 cos^{5} (x) d x

解

8

解答ステップ

\int_{0}^{\frac{π}{2}} 15 cos^{5} (x) d x

定数を除く:

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

= 15 \cdot \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

= 15 \cdot \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

= 15 \cdot \int_{0}^{1} (1 - u^{2})^{2} d u

拡張

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

= 15 \cdot \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

= 15 (\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}

= 15 (1 - \frac{2}{3} + \frac{1}{5})

簡素化

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

= 8

グラフ

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