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受欢迎的微积分 >

integral from 0 to pi/2 of 6cos^5(x)

解答

\int_{0}^{\frac{π}{2}} 6 cos^{5} (x) d x

解答

\frac{16}{5}

+1

十进制

3.2

求解步骤

\int_{0}^{\frac{π}{2}} 6 cos^{5} (x) d x

提出常数:

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

= 6 \cdot \int_{0}^{\frac{π}{2}} cos^{5} (x) d x

使用三角恒等式改写

= 6 \cdot \int_{0}^{\frac{π}{2}} (1 - sin^{2} (x))^{2} cos (x) d x

使用换元积分法

= 6 \cdot \int_{0}^{1} (1 - u^{2})^{2} d u

乘开

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

= 6 \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

= 6 (\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}

= 6 (1 - \frac{2}{3} + \frac{1}{5})

化简

6 (1 - \frac{2}{3} + \frac{1}{5}) : \frac{16}{5}

= \frac{16}{5}

作图

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