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受欢迎的微积分 >

integral from 0 to 0.5 of (1-x^2)^{0.5}

解答

\int_{0}^{0.5} (1 - x^{2})^{0.5} d x

解答

\frac{2 π + 3 3}{24}

+1

十进制

0.47830 \dots

求解步骤

\int_{0}^{0.5} (1 - x^{2})^{0.5} d x

化简

x^{2} : xx

= \int_{0}^{0.5} (1 - xx)^{0.5} d x

xx = x^{2}

= \int_{0}^{0.5} (1 - x^{2})^{\frac{1}{2}} d x

使用三角换元法

= \int_{0}^{\frac{π}{6}} cos^{2} (u) d u

使用三角恒等式改写

= \int_{0}^{\frac{π}{6}} \frac{1 + cos ( 2 u )}{2} d u

提出常数:

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

= \frac{1}{2} \cdot \int_{0}^{\frac{π}{6}} 1 + cos (2 u) d u

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_{0}^{\frac{π}{6}} 1 d u + \int_{0}^{\frac{π}{6}} cos (2 u) d u)

\int_{0}^{\frac{π}{6}} 1 d u = \frac{π}{6}

\int_{0}^{\frac{π}{6}} cos (2 u) d u = \frac{3}{4}

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

化简

\frac{1}{2} (\frac{π}{6} + \frac{3}{4}) : \frac{2 π + 3 3}{24}

= \frac{2 π + 3 3}{24}

作图

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