integral from-4pi to 3pi of cos^2(x)

解答

\int_{- 4 π}^{3 π} cos^{2} (x) d x

\frac{7 π}{2}

十进制

10.99557 \dots

求解步骤

\int_{- 4 π}^{3 π} cos^{2} (x) d x

使用三角恒等式改写

= \int_{- 4 π}^{3 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_{- 4 π}^{3 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_{- 4 π}^{3 π} 1 d x + \int_{- 4 π}^{3 π} cos (2 x) d x)

\int_{- 4 π}^{3 π} 1 d x = 7 π

\int_{- 4 π}^{3 π} cos (2 x) d x = 0

= \frac{1}{2} (7 π + 0)

化简

\frac{1}{2} (7 π + 0) : \frac{7 π}{2}

= \frac{7 π}{2}

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