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受欢迎的微积分 >

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

解答

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

解答

- \frac{16}{15}

+1

十进制

- 1.06666 \dots

求解步骤

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

提出常数:

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

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

使用三角恒等式改写

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

使用换元积分法

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

乘开

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

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

= - 2 (\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}

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

化简

- 2 (1 - \frac{2}{3} + \frac{1}{5}) : - \frac{16}{15}

= - \frac{16}{15}

作图

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