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受欢迎的微积分 >

integral from 0 to pi of sqrt(1+cos(2x))

解答

\int_{0}^{π} 1 + cos (2 x) d x

解答

22

+1

十进制

2.82842 \dots

求解步骤

\int_{0}^{π} 1 + cos (2 x) d x

使用三角恒等式改写

= \int_{0}^{π} 2 cos^{2} (x) d x

提出常数:

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

= 2 \cdot \int_{0}^{π} cos^{2} (x) d x

使用指数法则:

a^{2} = ∣ a ∣

= 2 \cdot \int_{0}^{π} ∣ cos (x) ∣ d x

拿掉绝对值

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

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

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

= 2 (1 + 1)

化简

= 22

作图

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