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受欢迎的微积分 >

integral from 0 to pi/2 of-6sin^2(3x)

解答

\int_{0}^{\frac{π}{2}} - 6 sin^{2} (3 x) d x

解答

- \frac{3 π}{2}

+1

十进制

- 4.71238 \dots

求解步骤

\int_{0}^{\frac{π}{2}} - 6 sin^{2} (3 x) d x

提出常数:

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

= - 6 \cdot \int_{0}^{\frac{π}{2}} sin^{2} (3 x) d x

使用三角恒等式改写

= - 6 \cdot \int_{0}^{\frac{π}{2}} \frac{1}{2} (1 - cos (6 x)) d x

提出常数:

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

= - 6 \cdot \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} 1 - cos (6 x) d x

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

= - 6 \cdot \frac{1}{2} (\int_{0}^{\frac{π}{2}} 1 d x - \int_{0}^{\frac{π}{2}} cos (6 x) d x)

\int_{0}^{\frac{π}{2}} 1 d x = \frac{π}{2}

\int_{0}^{\frac{π}{2}} cos (6 x) d x = 0

= - 6 \cdot \frac{1}{2} (\frac{π}{2} - 0)

化简

- 6 \cdot \frac{1}{2} (\frac{π}{2} - 0) : - \frac{3 π}{2}

= - \frac{3 π}{2}

作图

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