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受欢迎的微积分 >

integral from 0 to pi/3 of 5sin(x/2)

解答

\int_{0}^{\frac{π}{3}} 5 sin (\frac{x}{2}) d x

解答

10 (- \frac{3}{2} + 1)

+1

十进制

1.33974 \dots

求解步骤

\int_{0}^{\frac{π}{3}} 5 sin (\frac{x}{2}) d x

提出常数:

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

= 5 \cdot \int_{0}^{\frac{π}{3}} sin (\frac{x}{2}) d x

使用换元积分法

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

提出常数:

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

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

使用积分常见定则:

\int sin (u) d u = - cos (u)

= 5 \cdot 2 [- cos (u)]_{0}^{\frac{π}{6}}

化简

= 10 [- cos (u)]_{0}^{\frac{π}{6}}

带入上下限计算后得:

- \frac{3}{2} + 1

= 10 (- \frac{3}{2} + 1)

作图

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