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#1 2007-01-28 01:59:37

Toast
Real Member
Registered: 2006-10-08
Posts: 1,321

Integrating Trig

EDIT: I found the answer and I tried to delete this topic, but it wouldn't delete hmm lol


So I need to integrate this:

So I split up the product:


And simplify:

And integrate:

And simplify:

But the answer says:

I'm pretty certain I made a mistake with integrating cos(0), but I don't know how to fix it.

Last edited by Toast (2007-01-28 02:08:37)

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#2 2007-01-28 03:32:42

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Integrating Trig

Mmm, that happens to me sometimes as well. I bet if you tried to delete it now then it would work fine.

That first step you made is very nice, by the way. I haven't seen that method before, but it's very good.

I'd have done:
cos 2x = cos²x - sin²x
          = cos²x - (1 - cos²x)
          = 2cos²x - 1

∴ cos²x = (cos 2x + 1)/2
∴ cos²4x = (cos 8x + 1)/2.

Your method gets to the same place, but a lot quicker.

I'm sure you figured this out already, but your mistake was integrating cos(0) with respect to 0 instead of x. cos(0) = 1, and so integrating it will give x.


Why did the vector cross the road?
It wanted to be normal.

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#3 2007-01-28 16:35:55

Toast
Real Member
Registered: 2006-10-08
Posts: 1,321

Re: Integrating Trig

Just wondering, but what is the actual use for integrating trigonometry? You can't actually use it to find the area under a graph right, because the domain and range of all trigonometry functions are limited to all Real Numbers.

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#4 2007-01-28 21:44:07

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Integrating Trig

Of course you can! You just need to set boundaries on where the area you're interested in is, but that's the same for any time you want to find the area under a graph.

For example, say you wanted to know the area of one of the peaks that sin x has.
Integrate sin x between 0 and π, to get [-cos x] between 0 and π, which is 1 - (-1) = 2.
Very useful indeed.


Why did the vector cross the road?
It wanted to be normal.

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#5 2007-01-28 21:45:04

Dross
Member
Registered: 2006-08-24
Posts: 325

Re: Integrating Trig

Toast wrote:

You can't actually use it to find the area under a graph right, because the domain and range of all trigonometry functions are limited to all Real Numbers.


Why does that stop you finding the area under the graph? If you integrate cos(x) between 0 and
, you'd have the area under there. Some motion - particularly simple harmonic motion, which you should get to fairly soon if you're studying mechanics (and if you haven't already!) - is described using trigonometric functions.


Bad speling makes me [sic]

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#6 2007-01-28 22:36:51

Toast
Real Member
Registered: 2006-10-08
Posts: 1,321

Re: Integrating Trig

Oh, so you can only use radians when dealing with integrals? o_O I c.

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#7 2007-01-28 23:08:56

Dross
Member
Registered: 2006-08-24
Posts: 325

Re: Integrating Trig

Toast wrote:

Oh, so you can only use radians when dealing with integrals? o_O I c.

Well, you could use degrees, but the numbers wouldn't be as "nice". And certainly all the integration (and differentiation - remember to use radians there, too) formulae you've been taught that involve trigonometric functions will be in radians. Degrees are an artificial measurement of angle, whilst radians are more natrual.


Bad speling makes me [sic]

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