Integrate Cos 2a

Date: 08/15/2001 at 04:33:45
From: Katherine Watchorn
Subject: Integration of cos 2a

Dear Dr. Math,

I was doing some integration and one of the problems was to integrate
cos2a. The answer given in the book is sin 2a/2.

The textbooks that I have tell me that integrating cos a  gives me 
sin a, and tell me in detail how that was arrived at. But I have 
searched everywhere for information on cos2a and how the result 
sin 2a/2 is arrived at. Does this follow a pattern, i.e.   
cos 3a  = sin 3a/3  cos 4a = sin 4a/4   or  cos na  =  sin na/n  ?  

Many thanks for your help with previous questions; it was much 
appreciated.

Katherine Watchorn

Date: 08/15/2001 at 12:59:31
From: Doctor Peterson
Subject: Re: Integration of cos 2a

Hi, Katherine.

In general, the formula is

    [INT] cos(ax)dx = sin(ax) / a

You can prove this by taking the derivative of the right side:

    d/dx[1/a sin(ax)] = 1/a cos(ax) d/dx(ax)
                      = 1/a cos(ax) * a
                      = cos(ax)

This required only a simple application of the chain rule.

Not knowing this more general formula, you can obtain it by 
substitution. If we let u = ax, then du = a dx and dx = du/a, so

    [INT]cos(ax)dx = [INT]cos(u) du/a = sin(u)/a = sin(ax)/a

Here I first replaced ax with u and dx with du/a, then integrated, and 
finally replaced u with ax again.

The same method is useful everywhere, so you should learn it well and 
even be able to do it in your head.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/