What is the integral of cos(5x)sin(x)sin(8x) ?

The integral of cos(5x)sin(x)sin(8x) is

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integral of cos(5x)sin(x)sin(8x)

\int cos (5 x) sin (x) sin (8 x) d x

\frac{1}{2} cos (5 x) (- \frac{1}{9} sin (9 x) + \frac{1}{7} sin (7 x)) + \frac{5}{2} (- \frac{1}{18} (\frac{1}{4} sin (4 x) - \frac{1}{14} sin (14 x)) + \frac{1}{14} (\frac{1}{2} sin (2 x) - \frac{1}{12} sin (12 x))) + C

Solution steps

\int cos (5 x) sin (x) sin (8 x) d x

Apply Integration By Parts

= \frac{1}{2} cos (5 x) (- \frac{1}{9} sin (9 x) + \frac{1}{7} sin (7 x)) - \int - \frac{5}{2} sin (5 x) (- \frac{1}{9} sin (9 x) + \frac{1}{7} sin (7 x)) d x

\int - \frac{5}{2} sin (5 x) (- \frac{1}{9} sin (9 x) + \frac{1}{7} sin (7 x)) d x = - \frac{5}{2} (- \frac{1}{18} (\frac{1}{4} sin (4 x) - \frac{1}{14} sin (14 x)) + \frac{1}{14} (\frac{1}{2} sin (2 x) - \frac{1}{12} sin (12 x)))

= \frac{1}{2} cos (5 x) (- \frac{1}{9} sin (9 x) + \frac{1}{7} sin (7 x)) - (- \frac{5}{2} (- \frac{1}{18} (\frac{1}{4} sin (4 x) - \frac{1}{14} sin (14 x)) + \frac{1}{14} (\frac{1}{2} sin (2 x) - \frac{1}{12} sin (12 x))))

Simplify

= \frac{1}{2} cos (5 x) (- \frac{1}{9} sin (9 x) + \frac{1}{7} sin (7 x)) + \frac{5}{2} (- \frac{1}{18} (\frac{1}{4} sin (4 x) - \frac{1}{14} sin (14 x)) + \frac{1}{14} (\frac{1}{2} sin (2 x) - \frac{1}{12} sin (12 x)))

Add a constant to the solution

= \frac{1}{2} cos (5 x) (- \frac{1}{9} sin (9 x) + \frac{1}{7} sin (7 x)) + \frac{5}{2} (- \frac{1}{18} (\frac{1}{4} sin (4 x) - \frac{1}{14} sin (14 x)) + \frac{1}{14} (\frac{1}{2} sin (2 x) - \frac{1}{12} sin (12 x))) + C

What is the integral of cos(5x)sin(x)sin(8x) ?
The integral of cos(5x)sin(x)sin(8x) is