integral of sin(2x)sin(3x)cos(4x)

解答

\int sin (2 x) sin (3 x) cos (4 x) d x

\frac{1}{2} cos (4 x) sin (x) - \frac{1}{10} cos (4 x) sin (5 x) - \frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x) - \frac{1}{5} sin (x) + \frac{1}{45} sin (9 x) + C

求解步骤

\int sin (2 x) sin (3 x) cos (4 x) d x

使用分布积分法

= \frac{1}{2} cos (4 x) (sin (x) - \frac{1}{5} sin (5 x)) - \int - 2 sin (4 x) (sin (x) - \frac{1}{5} sin (5 x)) d x

\int - 2 sin (4 x) (sin (x) - \frac{1}{5} sin (5 x)) d x = \frac{1}{5} sin (5 x) - \frac{1}{3} sin (3 x) + \frac{1}{5} sin (x) - \frac{1}{45} sin (9 x)

= \frac{1}{2} cos (4 x) (sin (x) - \frac{1}{5} sin (5 x)) - (\frac{1}{5} sin (5 x) - \frac{1}{3} sin (3 x) + \frac{1}{5} sin (x) - \frac{1}{45} sin (9 x))

化简

\frac{1}{2} cos (4 x) (sin (x) - \frac{1}{5} sin (5 x)) - (\frac{1}{5} sin (5 x) - \frac{1}{3} sin (3 x) + \frac{1}{5} sin (x) - \frac{1}{45} sin (9 x)) : \frac{1}{2} cos (4 x) sin (x) - \frac{1}{10} cos (4 x) sin (5 x) - \frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x) - \frac{1}{5} sin (x) + \frac{1}{45} sin (9 x)

= \frac{1}{2} cos (4 x) sin (x) - \frac{1}{10} cos (4 x) sin (5 x) - \frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x) - \frac{1}{5} sin (x) + \frac{1}{45} sin (9 x)

解答补常数

= \frac{1}{2} cos (4 x) sin (x) - \frac{1}{10} cos (4 x) sin (5 x) - \frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x) - \frac{1}{5} sin (x) + \frac{1}{45} sin (9 x) + C

\int \frac{1}{x ^{2} + 3 x - 10} d x

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