integral from 0 to 1/2 of x*arcsin(x)

解答

\int_{0}^{\frac{1}{2}} x \cdot arcsin (x) d x

\frac{- π + 3 3}{48}

十进制

0.04280 \dots

求解步骤

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

使用分布积分法

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

\int \frac{x ^{2}}{2 1 - x ^{2}} d x = \frac{1}{4} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x)))

= [\frac{1}{2} (x^{2} arcsin (x) - 2 \cdot \frac{1}{4} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x))))]_{0}^{\frac{1}{2}}

2 \cdot \frac{1}{4} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x))) = \frac{1}{2} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x)))

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

带入上下限计算后得:

\frac{- π + 3 3}{48}

= \frac{- π + 3 3}{48}

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