integral from 0 to 1/2 of 3arccos(x)

解答

\int_{0}^{\frac{1}{2}} 3 arccos (x) d x

\frac{π - 3 3}{2} + 3

十进制

1.97272 \dots

求解步骤

\int_{0}^{\frac{1}{2}} 3 arccos (x) d x

提出常数:

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

= 3 \cdot \int_{0}^{\frac{1}{2}} arccos (x) d x

使用分布积分法

= 3 [x arccos (x) - \int - \frac{x}{1 - x ^{2}} d x]_{0}^{\frac{1}{2}}

\int - \frac{x}{1 - x ^{2}} d x = 1 - x^{2}

= 3 [x arccos (x) - 1 - x^{2}]_{0}^{\frac{1}{2}}

带入上下限计算后得:

\frac{π}{6} - \frac{3}{2} + 1

= 3 (\frac{π}{6} - \frac{3}{2} + 1)

化简

= \frac{π - 3 3}{2} + 3

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