integral of cos(x)cos(2x)sin(4x)

解答

\int cos (x) cos (2 x) sin (4 x) d x

- \frac{1}{10} cos (2 x) cos (5 x) - \frac{1}{6} cos (2 x) cos (3 x) + \frac{1}{70} cos (7 x) - \frac{1}{30} cos (3 x) + \frac{1}{30} cos (5 x) - \frac{1}{6} cos (x) + C

求解步骤

\int cos (x) cos (2 x) sin (4 x) d x

使用分布积分法

= \frac{1}{2} cos (2 x) (- \frac{1}{5} cos (5 x) - \frac{1}{3} cos (3 x)) - \int - sin (2 x) (- \frac{1}{5} cos (5 x) - \frac{1}{3} cos (3 x)) d x

\int - sin (2 x) (- \frac{1}{5} cos (5 x) - \frac{1}{3} cos (3 x)) d x = - \frac{1}{70} cos (7 x) + \frac{1}{30} cos (3 x) - \frac{1}{30} cos (5 x) + \frac{1}{6} cos (x)

= \frac{1}{2} cos (2 x) (- \frac{1}{5} cos (5 x) - \frac{1}{3} cos (3 x)) - (- \frac{1}{70} cos (7 x) + \frac{1}{30} cos (3 x) - \frac{1}{30} cos (5 x) + \frac{1}{6} cos (x))

化简

\frac{1}{2} cos (2 x) (- \frac{1}{5} cos (5 x) - \frac{1}{3} cos (3 x)) - (- \frac{1}{70} cos (7 x) + \frac{1}{30} cos (3 x) - \frac{1}{30} cos (5 x) + \frac{1}{6} cos (x)) : - \frac{1}{10} cos (2 x) cos (5 x) - \frac{1}{6} cos (2 x) cos (3 x) + \frac{1}{70} cos (7 x) - \frac{1}{30} cos (3 x) + \frac{1}{30} cos (5 x) - \frac{1}{6} cos (x)

= - \frac{1}{10} cos (2 x) cos (5 x) - \frac{1}{6} cos (2 x) cos (3 x) + \frac{1}{70} cos (7 x) - \frac{1}{30} cos (3 x) + \frac{1}{30} cos (5 x) - \frac{1}{6} cos (x)

解答补常数

= - \frac{1}{10} cos (2 x) cos (5 x) - \frac{1}{6} cos (2 x) cos (3 x) + \frac{1}{70} cos (7 x) - \frac{1}{30} cos (3 x) + \frac{1}{30} cos (5 x) - \frac{1}{6} cos (x) + C

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