integral from 0 to a of sin^2((npi)/a x)

解答

\int_{0}^{a} sin^{2} (\frac{nπ}{a} x) d x

\frac{a}{2} - \frac{a}{4 πn} sin (2 πn)

求解步骤

\int_{0}^{a} sin^{2} (\frac{nπ}{a} x) d x

使用换元积分法

= \int_{0}^{πn} \frac{a sin ^{2} ( u )}{πn} d u

提出常数:

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

= \frac{a}{πn} \cdot \int_{0}^{πn} sin^{2} (u) d u

使用三角恒等式改写

= \frac{a}{πn} \cdot \int_{0}^{πn} \frac{1 - cos ( 2 u )}{2} d u

提出常数:

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

= \frac{a}{πn} \cdot \frac{1}{2} \cdot \int_{0}^{πn} 1 - cos (2 u) d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{a}{πn} \cdot \frac{1}{2} (\int_{0}^{πn} 1 d u - \int_{0}^{πn} cos (2 u) d u)

\int_{0}^{πn} 1 d u = πn

\int_{0}^{πn} cos (2 u) d u = \frac{1}{2} sin (2 πn)

= \frac{a}{πn} \cdot \frac{1}{2} (πn - \frac{1}{2} sin (2 πn))

化简

\frac{a}{πn} \cdot \frac{1}{2} (πn - \frac{1}{2} sin (2 πn)) : \frac{a}{2} - \frac{a}{4 πn} sin (2 πn)

= \frac{a}{2} - \frac{a}{4 πn} sin (2 πn)

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