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受欢迎的微积分 >

integral from 0 to pi/8 of sin^2(2x)

解答

\int_{0}^{\frac{π}{8}} sin^{2} (2 x) d x

解答

\frac{π - 2}{16}

+1

十进制

0.07134 \dots

求解步骤

\int_{0}^{\frac{π}{8}} sin^{2} (2 x) d x

使用三角恒等式改写

= \int_{0}^{\frac{π}{8}} \frac{1}{2} (1 - cos (4 x)) d x

提出常数:

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

= \frac{1}{2} \cdot \int_{0}^{\frac{π}{8}} 1 - cos (4 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

= \frac{1}{2} (\int_{0}^{\frac{π}{8}} 1 d x - \int_{0}^{\frac{π}{8}} cos (4 x) d x)

\int_{0}^{\frac{π}{8}} 1 d x = \frac{π}{8}

\int_{0}^{\frac{π}{8}} cos (4 x) d x = \frac{1}{4}

= \frac{1}{2} (\frac{π}{8} - \frac{1}{4})

化简

\frac{1}{2} (\frac{π}{8} - \frac{1}{4}) : \frac{π - 2}{16}

= \frac{π - 2}{16}

作图

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