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受欢迎的微积分 >

integral from 0 to pi/8 of 3cos^5(4x)

解答

\int_{0}^{\frac{π}{8}} 3 cos^{5} (4 x) d x

解答

\frac{2}{5}

+1

十进制

0.4

求解步骤

\int_{0}^{\frac{π}{8}} 3 cos^{5} (4 x) d x

提出常数:

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

= 3 \cdot \int_{0}^{\frac{π}{8}} cos^{5} (4 x) d x

使用换元积分法

= 3 \cdot \int_{0}^{\frac{π}{2}} cos^{5} (u) \frac{1}{4} d u

提出常数:

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

= 3 \cdot \frac{1}{4} \cdot \int_{0}^{\frac{π}{2}} cos^{5} (u) d u

使用三角恒等式改写

= 3 \cdot \frac{1}{4} \cdot \int_{0}^{\frac{π}{2}} (1 - sin^{2} (u))^{2} cos (u) d u

使用换元积分法

= 3 \cdot \frac{1}{4} \cdot \int_{0}^{1} (1 - v^{2})^{2} d v

乘开

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

= 3 \cdot \frac{1}{4} \cdot \int_{0}^{1} 1 - 2 v^{2} + v^{4} d v

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

= 3 \cdot \frac{1}{4} (\int_{0}^{1} 1 d v - \int_{0}^{1} 2 v^{2} d v + \int_{0}^{1} v^{4} d v)

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

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

\int_{0}^{1} v^{4} d v = \frac{1}{5}

= 3 \cdot \frac{1}{4} (1 - \frac{2}{3} + \frac{1}{5})

化简

3 \cdot \frac{1}{4} (1 - \frac{2}{3} + \frac{1}{5}) : \frac{2}{5}

= \frac{2}{5}

作图

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