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

解答

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

\frac{π + 2}{16}

十进制

0.32134 \dots

求解步骤

\int_{0}^{\frac{π}{8}} cos^{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}