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受欢迎的微积分 >

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

解答

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

解答

- \frac{10}{3}

+1

十进制

- 3.33333 \dots

求解步骤

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

使用三角恒等式改写

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

提出常数:

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

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

使用换元积分法

= 5 (- \int_{0}^{π} sin (u) \frac{1}{3} d u)

提出常数:

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

= 5 (- \frac{1}{3} \cdot \int_{0}^{π} sin (u) d u)

使用积分常见定则:

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

= 5 (- \frac{1}{3} [- cos (u)]_{0}^{π})

化简

5 (- \frac{1}{3} [- cos (u)]_{0}^{π}) : - \frac{5}{3} [- cos (u)]_{0}^{π}

= - \frac{5}{3} [- cos (u)]_{0}^{π}

带入上下限计算后得:

2

= - \frac{5}{3} \cdot 2

化简

= - \frac{10}{3}

作图

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