integral from 0 to pi of (cos(θ))^2

解答

\int_{0}^{π} (cos (θ))^{2} d θ

\frac{π}{2}

十进制

1.57079 \dots

求解步骤

\int_{0}^{π} (cos (θ))^{2} d θ

化简

= \int_{0}^{π} cos^{2} (θ) d θ

使用三角恒等式改写

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

\int_{0}^{π} 1 d θ = π

\int_{0}^{π} cos (2 θ) d θ = 0

= \frac{1}{2} (π + 0)

化简

\frac{1}{2} (π + 0) : \frac{π}{2}

= \frac{π}{2}

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