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受欢迎的微积分 >

integral from 0 to pi/2 of cos^3(15x)

解答

\int_{0}^{\frac{π}{2}} cos^{3} (15 x) d x

解答

- \frac{2}{45}

+1

十进制

- 0.04444 \dots

求解步骤

\int_{0}^{\frac{π}{2}} cos^{3} (15 x) d x

使用换元积分法

= \int_{0}^{\frac{15 π}{2}} cos^{3} (u) \frac{1}{15} d u

提出常数:

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

= \frac{1}{15} \cdot \int_{0}^{\frac{15 π}{2}} cos^{3} (u) d u

使用三角恒等式改写

= \frac{1}{15} \cdot \int_{0}^{\frac{15 π}{2}} (1 - sin^{2} (u)) cos (u) d u

使用换元积分法

= \frac{1}{15} \cdot \int_{0}^{- 1} 1 - v^{2} d v

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= \frac{1}{15} (- \int_{- 1}^{0} 1 - v^{2} 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}{15} (- (\int_{- 1}^{0} 1 d v - \int_{- 1}^{0} v^{2} d v))

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

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

= \frac{1}{15} (- (1 - \frac{1}{3}))

化简

\frac{1}{15} (- (1 - \frac{1}{3})) : - \frac{2}{45}

= - \frac{2}{45}

作图

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