integral from 0 to 3/2 pi of xsin(2x)

解答

\int_{0}^{\frac{3}{2} π} x sin (2 x) d x

\frac{3 π}{4}

十进制

2.35619 \dots

求解步骤

\int_{0}^{\frac{3}{2} π} x sin (2 x) d x

使用分布积分法

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

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

= [\frac{1}{2} (- x cos (2 x) - 2 (- \frac{1}{4} sin (2 x)))]_{0}^{π \frac{3}{2}}

化简

[\frac{1}{2} (- x cos (2 x) - 2 (- \frac{1}{4} sin (2 x)))]_{0}^{π \frac{3}{2}} : [\frac{1}{2} (- x cos (2 x) + \frac{1}{2} sin (2 x))]_{0}^{π \frac{3}{2}}

= [\frac{1}{2} (- x cos (2 x) + \frac{1}{2} sin (2 x))]_{0}^{π \frac{3}{2}}

带入上下限计算后得:

\frac{3 π}{4}

= \frac{3 π}{4}

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