integral from 0 to pi/4 of cos^2(x/3)

解答

\int_{0}^{\frac{π}{4}} cos^{2} (\frac{x}{3}) d x

\frac{π + 3}{8}

十进制

0.76769 \dots

求解步骤

\int_{0}^{\frac{π}{4}} cos^{2} (\frac{x}{3}) d x

使用三角恒等式改写

= \int_{0}^{\frac{π}{4}} \frac{1}{2} (1 + cos (\frac{2 x}{3})) 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}{4}} 1 + cos (\frac{2 x}{3}) 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}{4}} 1 d x + \int_{0}^{\frac{π}{4}} cos (\frac{2 x}{3}) d x)

\int_{0}^{\frac{π}{4}} 1 d x = \frac{π}{4}

\int_{0}^{\frac{π}{4}} cos (\frac{2 x}{3}) d x = \frac{3}{4}

= \frac{1}{2} (\frac{π}{4} + \frac{3}{4})

化简

\frac{1}{2} (\frac{π}{4} + \frac{3}{4}) : \frac{π + 3}{8}

= \frac{π + 3}{8}

\int_{1}^{2} (x^{2} + 2 x) d x

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