integral from 0 to pi/3 of cos^2(θ)

解答

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

\frac{4 π + 3 3}{24}

十进制

0.74010 \dots

求解步骤

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

使用三角恒等式改写

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

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}{3}} 1 + cos (2 θ) d θ

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

\int_{0}^{\frac{π}{3}} 1 d θ = \frac{π}{3}

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

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

化简

\frac{1}{2} (\frac{π}{3} + \frac{3}{4}) : \frac{4 π + 3 3}{24}

= \frac{4 π + 3 3}{24}

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