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受欢迎的微积分 >

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

解答

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

解答

\frac{π ^{2}}{4}

+1

十进制

2.46740 \dots

求解步骤

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

提出常数:

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

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

使用三角恒等式改写

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

提出常数:

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

= π \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} 1 + cos (2 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

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

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

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

= π \frac{1}{2} (\frac{π}{2} + 0)

化简

π \frac{1}{2} (\frac{π}{2} + 0) : \frac{π ^{2}}{4}

= \frac{π ^{2}}{4}

作图

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