integral from 0 to 2pi of θcos(nθ)

解答

\int_{0}^{2 π} θ cos (n θ) d θ

\frac{2 π sin ( 2 πn )}{n}

求解步骤

\int_{0}^{2 π} θ cos (n θ) d θ

使用换元积分法

= \int_{0}^{n \cdot 2 π} \frac{u cos ( u )}{n ^{2}} d u

提出常数:

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

= \frac{1}{n ^{2}} \cdot \int_{0}^{n \cdot 2 π} u cos (u) d u

使用分布积分法

= \frac{1}{n ^{2}} [u sin (u) - \int sin (u) d u]_{0}^{2 πn}

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

= \frac{1}{n ^{2}} [u sin (u) - (- cos (u))]_{0}^{2 πn}

化简

= \frac{1}{n ^{2}} [u sin (u) + cos (u)]_{0}^{2 πn}

带入上下限计算后得:

2 πn sin (2 πn)

= \frac{1}{n ^{2}} \cdot 2 πn sin (2 πn)

化简

= \frac{2 π sin ( 2 πn )}{n}

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