integral from-pi to pi of xsin(nx)

解答

\int_{- π}^{π} x \cdot sin (n \cdot x) d x

- \frac{2 π ( - 1 ) ^{n}}{n}

求解步骤

\int_{- π}^{π} x sin (n x) d x

使用分布积分法

= [\frac{1}{n} (- x cos (n x) - n \cdot \int - \frac{1}{n} cos (n x) d x)]_{- π}^{π}

\int - \frac{1}{n} cos (n x) d x = - \frac{1}{n ^{2}} sin (n x)

= [\frac{1}{n} (- x cos (n x) - n (- \frac{1}{n ^{2}} sin (n x)))]_{- π}^{π}

化简

[\frac{1}{n} (- x cos (n x) - n (- \frac{1}{n ^{2}} sin (n x)))]_{- π}^{π} : [\frac{1}{n} (- x cos (n x) + \frac{1}{n} sin (n x))]_{- π}^{π}

= [\frac{1}{n} (- x cos (n x) + \frac{1}{n} sin (n x))]_{- π}^{π}

带入上下限计算后得:

- 2 (- 1)^{n} \frac{π}{n}

= - 2 (- 1)^{n} \frac{π}{n}

化简

= - \frac{2 π ( - 1 ) ^{n}}{n}

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