integral from 0 to 4 of sin((npix)/4)

解答

\int_{0}^{4} sin (\frac{nπ x}{4}) d x

\frac{4 ( - ( - 1 ) ^{n} + 1 )}{πn}

求解步骤

\int_{0}^{4} sin (\frac{nπ x}{4}) d x

使用换元积分法

= \int_{0}^{πn} \frac{4 sin ( u )}{πn} d u

提出常数:

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

= \frac{4}{πn} \cdot \int_{0}^{πn} sin (u) d u

使用积分常见定则:

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

= \frac{4}{πn} [- cos (u)]_{0}^{πn}

带入上下限计算后得:

- (- 1)^{n} + 1

= \frac{4}{πn} (- (- 1)^{n} + 1)

化简

= \frac{4 ( - ( - 1 ) ^{n} + 1 )}{πn}

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