Math Question

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I’ve been trying forever to suss out this problem (I suspect a derivative of it will appear on a quiz tomorrow), and nobody I know can work it out either, so if someone here could kindly give me some guidance, it’d be much appreciated. I’ve typed up the problem and the work I’ve done so far on it:



So, I’m stuck at the end, since it should come out without any imaginary numbers. I suspect I’m missing something really simple and obvious, but my brain’s fried right now. I’ve also tried substituting a couple double-angle identities into cos(2x), but those led to dead-ends too.
 
I was about to hack away at it using integration by parts until I saw the i.

As ashamed as I am to not recognize it, would you care to explain what it is exactly?

Also, have you by chance happened to have studied Taylor Series and Taylor Polynomials?
 
Just browsing over it quickly, I saw nothing wrong with what you had done there - though, it's been years since I've done any calculus with complex numbers.
My only thought to use the Euler equation to get rid of the e^7ix and replace it with some sin's and cos's in your first step. But doing that makes the integration 10x more messy (requires substitution), and I didn't feel like pursuing the idea any further.

I plugged the question into MATLAB, and it spewed forth the following:

Code: 
>> x = sym ('x')
 
x =
 
x
 
 
>> f=exp(i*7*x)*cos(2*x)
 
f =
 
exp(7*i*x)*cos(2*x)
 
 
>> int(f,x)
 
ans =
 
-7/45*i*exp(7*i*x)*cos(2*x)-2/45*exp(7*i*x)*sin(2*x)

that is:

complex_exponential_integral.JPG


Sorry, that's probably of no help. I realize it still has complex exponentials in it, and you think it shouldn't.
As I said, it's been 3 years since I've done anything with complex numbers and I wouldn't trust myself too much to try and give you a answer I got by hand.

Sorry :indiff: Hopefully someone else has something more constructive for you.


edit:
I found an identity in the back of my calculus text

integral_identity.JPG


Using that identity gives you precisely what I already posted.
 
dougiemeats
As ashamed as I am to not recognize it, would you care to explain what it is exactly?
Click to expand...
Well, the first step I show is a direct substitution for the exponential form of cosine (the stuff in brackets equals cos2x), and supposedly doing this substitution can make certain integration problems easier. The steps I’ve done after that are all algebra and regular integration.

Also, have you by chance happened to have studied Taylor Series and Taylor Polynomials?
Click to expand...
Nope.

Boundary Layer
Sorry, that's probably of no help. I realize it still has complex exponentials in it, and you think it shouldn't.
Click to expand...
You know what, now that I think about it, maybe I’m making the wrong assumption – I just now realized that since the original problem has i in it, maybe the answer is allowed to have it too. My professor was stressing that our answers needed to be real, but maybe he’d forgotten about that one problem…

Well darn, now I’m not even sure if I’m wrong. :boggled:

Thanks anyway for the help to both of you! :)

[edit]: Hmmm, it looks like my answer might be leading to what you just posted. I’ll try hacking away at it a bit.
 
Sage
You know what, now that I think about it, maybe I’m making the wrong assumption – I just now realized that since the original problem has i in it, maybe the answer is allowed to have it too. My professor was stressing that our answers needed to be real, but maybe he’d forgotten about that one problem…

Well darn, now I’m not even sure if I’m wrong. :boggled:
Click to expand...

heh. I had figured you knew the form of the answer or something like that when you stated that. The assumption made no sense to me either but I didn't want to second guess you given my lack of recent exposure to this variety of problems. I think your only shot at getting rid of the i is through the Euler equation if you have multiples of pi in the exponent.

I've dug up some old notes of mine on complex calculus and Maclaurin series. I'll try working through your problem again and will post if I make any progress. For now though, I'd say the identity I editted in to my previous post is about as simplified an answer as you can get.
 
Okay, I went through with that idea I had before about replacing e^7ix by cos(7x) + sin(7ix) and then doing the integration.

Here's what I got:

complex_exponential_integral2.JPG

...the form looks correct for integrating cos(a)*cos(b) + sin(a)*cos(b), so to the best of my knowledge, this answer is correct...

I haven't checked to see that this format is equivalent to the other one I posted that keeps the exponentials. I'm gonna try to do that know, but my eyelids are started to feel heavy.
 
Ah, I think you’ve nailed it! I totally forgot about doing that substitution.

Whether or not it turns out to be right, thank you very very very very much. :)👍 I owe you one.
 
I think you need to use integration by parts here twice.

Working it out now...

EDIT: I'm either too sloppy or wrong, and I have to go. Meh.

EDIT2: Found the error, and I got -4/45 * (e^(i7x)/2 * sin(2x) + 7ie^(i7x)/4 * cos(2x))

That actually looks okay, although I think I messed up some arithmetic. Anyways I used u = e^i7x and dv = cos 2x dx then u = e^i7x and dv = sin 2x dx then put all the integrals on the left.
 

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