integral from-pi to pi of xsin(mx)

解

\int_{- π}^{π} x sin (m x) d x

- \frac{2 π cos ( πm )}{m} + \frac{2 sin ( πm )}{m ^{2}}

解答ステップ

\int_{- π}^{π} x sin (m x) d x

部分積分を適用する

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

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

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

簡素化

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

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

限度を計算する:

- 2 \cdot \frac{π}{m} cos (πm) + \frac{2}{m ^{2}} sin (πm)

= - 2 \cdot \frac{π}{m} cos (πm) + \frac{2}{m ^{2}} sin (πm)

簡素化

= - \frac{2 π cos ( πm )}{m} + \frac{2 sin ( πm )}{m ^{2}}