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

解答

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

\frac{π}{4} + \frac{1}{2}

十进制

1.28539 \dots

求解步骤

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

使用三角恒等式改写

= 2 \cdot \int_{0}^{\frac{π}{4}} \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

= 2 \cdot \frac{1}{2} \cdot \int_{0}^{\frac{π}{4}} 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

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

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

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

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

化简

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

= \frac{π}{4} + \frac{1}{2}

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