Higher Level Integration Marathon & Questions (2 Viewers)

InteGrand

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Re: Extracurricular Integration Marathon

Here's another nice question.

 

omegadot

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Re: Extracurricular Integration Marathon

For 4, I think people should still attempt it / try other methods, as it is much less clear why "letting a=-i" should be valid. Letting a=-1 would give us something nonsensical for example. Definitely need to say something more to justify the formula being the same as that of the Gaussian integral (and why that particular choice of square root and not the other).
























 

seanieg89

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Re: Extracurricular Integration Marathon

Nice :), it's a bit lengthier than you can do it using complex analysis but I like the fact that a high school student could understand it.
 

InteGrand

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Re: Extracurricular Integration Marathon

The good old tedious integral of 1/(1+t^4) haha.
 

InteGrand

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Re: Extracurricular Integration Marathon

Ah right, I was thinking of the one with a t2 on the numerator too (the one that comes about when doing integral of √(tan x) ). That one was more tedious iirc. (Split into partial fractions etc.)
 

Paradoxica

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Re: Extracurricular Integration Marathon

Ah right, I was thinking of the one with a t2 on the numerator too (the one that comes about when doing integral of √(tan x) ). That one was more tedious iirc. (Split into partial fractions etc.)
No, that one is amenable to double symmetric substitution as well.

 
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RealiseNothing

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Re: Extracurricular Integration Marathon

Let for

Then

So we have:





To make things neater we will define from now on

We then integrate counter-clockwise about a boundary formed by to make a circular arc and get:



Now take the limit as



Now from a previously answered integral we know that:



So we then have:



Using Euler's formula we obtain:



Since the imaginary part = 0 we get:



Now equating the real part gives:







Then we just double the answer cos even function to get:

 
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seanieg89

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Re: Extracurricular Integration Marathon

Excellent solution to the integral :).

One minor remark is that another way of proving the summation result used is taking the real part of the geometric series summation (common ratio z=a*cis(x)).

I.e. find the real part of 1/(1-a*cis(x)), which is quite quick.
 

omegadot

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Re: Extracurricular Integration Marathon

Excellent solution to the integral :).

One minor remark is that another way of proving the summation result used is taking the real part of the geometric series summation (common ratio z=a*cis(x)).

I.e. find the real part of 1/(1-a*cis(x)), which is quite quick.
Yes, I know, but I was trying to keep the problem "purely real" for the benefit of any MX2 students who may care to read this thread (though those who are reading this thread probably already know or could follow the complex way anyway).
 

Paradoxica

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Re: Extracurricular Integration Marathon

Yes, I know, but I was trying to keep the problem "purely real" for the benefit of any MX2 students who may care to read this thread (though those who are reading this thread probably already know or could follow the complex way anyway).
Don't bother, anyone who can follow this thread on the real analysis side probably has what it takes to follow the complex analysis side.

Anyway, nice answer. Here's the one I got from the place I found the problem.



 

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