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### Integrate Cos 2a

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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

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Calculus

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