integral from y to 2 of sin((pix)/(2y))

解答

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

- \frac{2 y cos ( \frac{π}{y} )}{π}

求解步骤

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

使用换元积分法

= \int_{\frac{π}{2}}^{\frac{π}{y}} \frac{2 y sin ( u )}{π} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{2 y}{π} \cdot \int_{\frac{π}{2}}^{\frac{π}{y}} sin (u) d u

使用积分常见定则:

\int sin (u) d u = - cos (u)

= \frac{2 y}{π} [- cos (u)]_{\frac{π}{2}}^{\frac{π}{y}}

带入上下限计算后得:

- cos (\frac{π}{y})

= \frac{2 y}{π} (- cos (\frac{π}{y}))

化简

= - \frac{2 y cos ( \frac{π}{y} )}{π}

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