integral from-pi to pi of xsin(ax)

解答

\int_{- π}^{π} x sin (a x) d x

- \frac{2 π cos ( πa )}{a} + \frac{2 sin ( πa )}{a ^{2}}

求解步骤

\int_{- π}^{π} x sin (a x) d x

使用分布积分法

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

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

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

化简

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

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

带入上下限计算后得:

- 2 \cdot \frac{π}{a} cos (πa) + \frac{2}{a ^{2}} sin (πa)

= - 2 \cdot \frac{π}{a} cos (πa) + \frac{2}{a ^{2}} sin (πa)

化简

= - \frac{2 π cos ( πa )}{a} + \frac{2 sin ( πa )}{a ^{2}}

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