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受欢迎的微积分 >

integral from-1 to 1 of-2cos(pi/3 x)

解答

\int_{- 1}^{1} - 2 cos (\frac{π}{3} x) d x

解答

- \frac{6 3}{π}

+1

十进制

- 3.30797 \dots

求解步骤

\int_{- 1}^{1} - 2 cos (\frac{π}{3} x) d x

提出常数:

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

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

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

使用换元积分法

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

提出常数:

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

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

使用积分常见定则:

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

= - 2 \cdot \frac{3}{π} [sin (u)]_{- \frac{π}{3}}^{\frac{π}{3}}

化简

- 2 \cdot \frac{3}{π} [sin (u)]_{- \frac{π}{3}}^{\frac{π}{3}} : - \frac{6}{π} [sin (u)]_{- \frac{π}{3}}^{\frac{π}{3}}

= - \frac{6}{π} [sin (u)]_{- \frac{π}{3}}^{\frac{π}{3}}

带入上下限计算后得:

3

= - \frac{6}{π} 3

化简

= - \frac{6 3}{π}

作图

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