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人気のある微分積分 >

integral from 0 to pi/2 of cos^{11}(x)

解

\int_{0}^{\frac{π}{2}} cos^{11} (x) d x

解

\frac{256}{693}

+1

十進法表記

0.36940 \dots

解答ステップ

\int_{0}^{\frac{π}{2}} cos^{11} (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))^{5} cos (x) d x

u置換積分法を適用する

= \int_{0}^{1} (1 - u^{2})^{5} d u

拡張

(1 - u^{2})^{5} : 1 - 5 u^{2} + 10 u^{4} - 10 u^{6} + 5 u^{8} - u^{10}

= \int_{0}^{1} 1 - 5 u^{2} + 10 u^{4} - 10 u^{6} + 5 u^{8} - u^{10} 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} 5 u^{2} d u + \int_{0}^{1} 10 u^{4} d u - \int_{0}^{1} 10 u^{6} d u + \int_{0}^{1} 5 u^{8} d u - \int_{0}^{1} u^{10} d u

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

\int_{0}^{1} 5 u^{2} d u = \frac{5}{3}

\int_{0}^{1} 10 u^{4} d u = 2

\int_{0}^{1} 10 u^{6} d u = \frac{10}{7}

\int_{0}^{1} 5 u^{8} d u = \frac{5}{9}

\int_{0}^{1} u^{10} d u = \frac{1}{11}

= 1 - \frac{5}{3} + 2 - \frac{10}{7} + \frac{5}{9} - \frac{1}{11}

簡素化

1 - \frac{5}{3} + 2 - \frac{10}{7} + \frac{5}{9} - \frac{1}{11} : \frac{256}{693}

= \frac{256}{693}

グラフ

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