integral from 0 to L of cos((npix)/L)

解答

\int_{0}^{L} cos (\frac{nπ x}{L}) d x

0

求解步骤

\int_{0}^{L} cos (\frac{nπ x}{L}) d x

使用换元积分法

= \int_{0}^{n L π} \frac{cos ( \frac{u}{L} )}{πn} d u

提出常数:

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

= \frac{1}{πn} \cdot \int_{0}^{n L π} cos (\frac{u}{L}) d u

使用换元积分法

= \frac{1}{πn} \cdot \int_{0}^{πn} cos (v) L d v

提出常数:

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

= \frac{1}{πn} L \cdot \int_{0}^{πn} cos (v) d v

使用积分常见定则:

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

= \frac{1}{πn} L [sin (v)]_{0}^{πn}

带入上下限计算后得:

0

= \frac{1}{πn} L \cdot 0

化简

= 0

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