integral from-H to H of cos((ypi)/(2H))

解答

\int_{- H}^{H} cos (\frac{y π}{2 H}) d y

\frac{4 H}{π}

求解步骤

\int_{- H}^{H} cos (\frac{y π}{2 H}) d y

使用换元积分法

= \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{2 H cos ( u )}{π} d u

提出常数:

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

= \frac{2 H}{π} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} cos (u) d u

使用积分常见定则:

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

= \frac{2 H}{π} [sin (u)]_{- \frac{π}{2}}^{\frac{π}{2}}

带入上下限计算后得:

2

= \frac{2 H}{π} \cdot 2

化简

= \frac{4 H}{π}

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