integral from-pi to 2pi of cos^2(x)

解答

\int_{- π}^{2 π} cos^{2} (x) d x

\frac{3 π}{2}

十进制

4.71238 \dots

求解步骤

\int_{- π}^{2 π} cos^{2} (x) d x

使用三角恒等式改写

= \int_{- π}^{2 π} \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

= \frac{1}{2} \cdot \int_{- π}^{2 π} 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

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

\int_{- π}^{2 π} 1 d x = 3 π

\int_{- π}^{2 π} cos (2 x) d x = 0

= \frac{1}{2} (3 π + 0)

化简

\frac{1}{2} (3 π + 0) : \frac{3 π}{2}

= \frac{3 π}{2}

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