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

解答

\int_{\frac{π}{3}}^{\frac{π}{2}} 3 cos^{2} (x) d x

\frac{2 π - 3 3}{8}

十进制

0.13587 \dots

求解步骤

\int_{\frac{π}{3}}^{\frac{π}{2}} 3 cos^{2} (x) d x

提出常数:

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

= 3 \cdot \int_{\frac{π}{3}}^{\frac{π}{2}} cos^{2} (x) d x

使用三角恒等式改写

= 3 \cdot \int_{\frac{π}{3}}^{\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

= 3 \cdot \frac{1}{2} \cdot \int_{\frac{π}{3}}^{\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

= 3 \cdot \frac{1}{2} (\int_{\frac{π}{3}}^{\frac{π}{2}} 1 d x + \int_{\frac{π}{3}}^{\frac{π}{2}} cos (2 x) d x)

\int_{\frac{π}{3}}^{\frac{π}{2}} 1 d x = \frac{π}{6}

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

= 3 \cdot \frac{1}{2} (\frac{π}{6} - \frac{3}{4})

化简

3 \cdot \frac{1}{2} (\frac{π}{6} - \frac{3}{4}) : \frac{2 π - 3 3}{8}

= \frac{2 π - 3 3}{8}

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