integral from 0 to 0.5 of arcsin(x)

解答

\int_{0}^{0.5} arcsin (x) d x

0.12782 \dots

求解步骤

\int_{0}^{0.5} arcsin (x) d x

使用分布积分法

= [x arcsin (x) - \int \frac{x}{1 - x ^{2}} d x]_{0}^{0.5}

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

= [x arcsin (x) - (- 1 - x^{2})]_{0}^{0.5}

化简

= [x arcsin (x) + 1 - x^{2}]_{0}^{0.5}

带入上下限计算后得:

0.12782 \dots

= 0.12782 \dots

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