integral from-pi to pi of sin(nx)sin(mx)

解

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

- \frac{1}{n + m} sin (πn + πm) + \frac{sin ( πn - πm ) - sin ( - πn + πm )}{2 ( n - m )}

解答ステップ

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

三角関数の公式を使用して書き換える

= \int_{- π}^{π} \frac{- cos ( n x + m x ) + cos ( n x - m x )}{2} d x

定数を除く:

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

= \frac{1}{2} \cdot \int_{- π}^{π} - cos (n x + m x) + cos (n x - m x) d x

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (- \int_{- π}^{π} cos (n x + m x) d x + \int_{- π}^{π} cos (n x - m x) d x)

\int_{- π}^{π} cos (n x + m x) d x = \frac{2 sin ( πn + πm )}{n + m}

\int_{- π}^{π} cos (n x - m x) d x = \frac{sin ( πn - πm ) - sin ( - πn + πm )}{n - m}

= \frac{1}{2} (- \frac{2 sin ( πn + πm )}{n + m} + \frac{sin ( πn - πm ) - sin ( - πn + πm )}{n - m})

簡素化

\frac{1}{2} (- \frac{2 sin ( πn + πm )}{n + m} + \frac{sin ( πn - πm ) - sin ( - πn + πm )}{n - m}) : - \frac{1}{n + m} sin (πn + πm) + \frac{sin ( πn - πm ) - sin ( - πn + πm )}{2 ( n - m )}

= - \frac{1}{n + m} sin (πn + πm) + \frac{sin ( πn - πm ) - sin ( - πn + πm )}{2 ( n - m )}