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

解答

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

\frac{256}{693}

十进制

0.36940 \dots

求解步骤

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

使用三角恒等式改写

= \int_{0}^{\frac{π}{2}} (1 - sin^{2} (x))^{5} cos (x) d x

使用换元积分法

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

乘开

(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

使用积分加法定则:

\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}

化简

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

= \frac{256}{693}

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