What is the integral of cos(x)cos(3x)cos(5x) ?

The integral of cos(x)cos(3x)cos(5x) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of cos(x)cos(3x)cos(5x)

\int cos (x) cos (3 x) cos (5 x) d x

\frac{1}{2} cos (x) (\frac{1}{2} sin (2 x) + \frac{1}{8} sin (8 x)) + \frac{1}{2} (\frac{1}{3} sin^{3} (x) + \frac{1}{16} (\frac{1}{7} sin (7 x) - \frac{1}{9} sin (9 x))) + C

Solution steps

\int cos (x) cos (3 x) cos (5 x) d x

Apply Integration By Parts

= \frac{1}{2} cos (x) (\frac{1}{2} sin (2 x) + \frac{1}{8} sin (8 x)) - \int - \frac{1}{2} sin (x) (\frac{1}{2} sin (2 x) + \frac{1}{8} sin (8 x)) d x

\int - \frac{1}{2} sin (x) (\frac{1}{2} sin (2 x) + \frac{1}{8} sin (8 x)) d x = - \frac{1}{2} (\frac{1}{3} sin^{3} (x) + \frac{1}{16} (\frac{1}{7} sin (7 x) - \frac{1}{9} sin (9 x)))

= \frac{1}{2} cos (x) (\frac{1}{2} sin (2 x) + \frac{1}{8} sin (8 x)) - (- \frac{1}{2} (\frac{1}{3} sin^{3} (x) + \frac{1}{16} (\frac{1}{7} sin (7 x) - \frac{1}{9} sin (9 x))))

Simplify

= \frac{1}{2} cos (x) (\frac{1}{2} sin (2 x) + \frac{1}{8} sin (8 x)) + \frac{1}{2} (\frac{1}{3} sin^{3} (x) + \frac{1}{16} (\frac{1}{7} sin (7 x) - \frac{1}{9} sin (9 x)))

Add a constant to the solution

= \frac{1}{2} cos (x) (\frac{1}{2} sin (2 x) + \frac{1}{8} sin (8 x)) + \frac{1}{2} (\frac{1}{3} sin^{3} (x) + \frac{1}{16} (\frac{1}{7} sin (7 x) - \frac{1}{9} sin (9 x))) + C

What is the integral of cos(x)cos(3x)cos(5x) ?
The integral of cos(x)cos(3x)cos(5x) is