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受欢迎的微积分 >

integral from 0 to pi/6 of cos^7(3x)

解答

\int_{0}^{\frac{π}{6}} cos^{7} (3 x) d x

解答

\frac{16}{105}

+1

十进制

0.15238 \dots

求解步骤

\int_{0}^{\frac{π}{6}} cos^{7} (3 x) d x

使用换元积分法

= \int_{0}^{\frac{π}{2}} cos^{7} (u) \frac{1}{3} d u

提出常数:

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

= \frac{1}{3} \cdot \int_{0}^{\frac{π}{2}} cos^{7} (u) d u

使用三角恒等式改写

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

使用换元积分法

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

乘开

(1 - v^{2})^{3} : 1 - 3 v^{2} + 3 v^{4} - v^{6}

= \frac{1}{3} \cdot \int_{0}^{1} 1 - 3 v^{2} + 3 v^{4} - v^{6} 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

= \frac{1}{3} (\int_{0}^{1} 1 d v - \int_{0}^{1} 3 v^{2} d v + \int_{0}^{1} 3 v^{4} d v - \int_{0}^{1} v^{6} d v)

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

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

\int_{0}^{1} 3 v^{4} d v = \frac{3}{5}

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

= \frac{1}{3} (1 - 1 + \frac{3}{5} - \frac{1}{7})

化简

\frac{1}{3} (1 - 1 + \frac{3}{5} - \frac{1}{7}) : \frac{16}{105}

= \frac{16}{105}

作图

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