integral from 0 to 1/2 of arcsin(y)

解答

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

\frac{π}{12} + \frac{3}{2} - 1

十进制

0.12782 \dots

求解步骤

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

使用分布积分法

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

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

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

化简

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

带入上下限计算后得:

\frac{π}{12} + \frac{3}{2} - 1

= \frac{π}{12} + \frac{3}{2} - 1

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