integral from 0 to pi/6 of xcos(x+pi/3)

解答

\int_{0}^{\frac{π}{6}} x cos (x + \frac{π}{3}) d x

\frac{π}{6} - \frac{1}{2}

十进制

0.02359 \dots

求解步骤

\int_{0}^{\frac{π}{6}} x cos (x + \frac{π}{3}) d x

使用分布积分法

= [x sin (x + \frac{π}{3}) - \int sin (x + \frac{π}{3}) d x]_{0}^{\frac{π}{6}}

\int sin (x + \frac{π}{3}) d x = - cos (x + \frac{π}{3})

= [x sin (x + \frac{π}{3}) - (- cos (x + \frac{π}{3}))]_{0}^{\frac{π}{6}}

化简

= [x sin (x + \frac{π}{3}) + cos (x + \frac{π}{3})]_{0}^{\frac{π}{6}}

带入上下限计算后得:

\frac{π}{6} - \frac{1}{2}

= \frac{π}{6} - \frac{1}{2}

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