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Thread: Higher Level Integration Marathon & Questions

  1. #26
    Rambling Spirit
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    Re: Extracurricular Integration Marathon

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

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    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by InteGrand View Post
    The good old tedious integral of 1/(1+t^4) haha.
    Last edited by Paradoxica; 31 Jan 2016 at 11:17 PM.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  3. #28
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post
    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.)

  4. #29
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by InteGrand View Post
    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.

    Last edited by Paradoxica; 9 Jan 2016 at 1:24 PM.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  5. #30
    what is that?It is Cowpea RealiseNothing's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by seanieg89 View Post
    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:

    Last edited by RealiseNothing; 9 Jan 2016 at 4:25 AM.

  6. #31
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  7. #32
    Executive Member KingOfActing's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post
    I've seen the solution to that integral before, good luck everyone.
    Spoiler (rollover to view):
    3 substitutions and residue theorem
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  8. #33
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by KingOfActing View Post
    I've seen the solution to that integral before, good luck everyone.
    Is that the most straightforward way to do it?

    Surely there must be more straightforward ways.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  9. #34
    Executive Member KingOfActing's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post
    Is that the most straightforward way to do it?

    Surely there must be more straightforward ways.
    It's the way I've seen it done, there could be a simpler way
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  10. #35
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post
    Here's a very deceptive integral which appears simple.


  11. #36
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post
    Here's a very deceptive integral which appears simple.












































  12. #37
    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by omegadot View Post
    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.

  13. #38
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by seanieg89 View Post
    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).

  14. #39
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by omegadot View Post
    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.



    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Integration Marathon


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    Ancient Orator leehuan's Avatar
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    Re: Extracurricular Integration Marathon

    Extra question:

    Last edited by leehuan; 17 Jan 2016 at 11:09 PM.

  17. #42
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by seanieg89 View Post










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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by leehuan View Post
    Extra question:



    Last edited by omegadot; 24 Feb 2017 at 5:05 PM. Reason: Typo
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by InteGrand View Post
    Here's another nice question.

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    Re: Extracurricular Integration Marathon




  21. #46
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    Re: Extracurricular Integration Marathon

    Paradoxica and omegadot like this.

  22. #47
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by seanieg89 View Post
    Beat me to it by nine hours But my solution involved factorising out the 1/4 so basel's could be seen easily. But my main problem is that I can't get rid of the negative that appears in the series expansion for log(1-x). Here's my solution, which ignores the sign error...



    WHERE DID THE MISSING SIGN GO?!?!?!
    Last edited by Paradoxica; 20 Jan 2016 at 8:58 PM.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  23. #48
    Loquacious One Drsoccerball's Avatar
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    Re: Extracurricular Integration Marathon






  24. #49
    Rambling Spirit
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post
    Beat me to it by nine hours But my solution involved factorising out the 1/4 so basel's could be seen easily. But my main problem is that I can't get rid of the negative that appears in the series expansion for log(1-x). Here's my solution, which ignores the sign error...



    WHERE DID THE MISSING SIGN GO?!?!?!

  25. #50
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Drsoccerball View Post




    Minkowski Inequality easily destroys the odd valued cases, but is itself destroyed by the even valued cases.

    I hope that's what you meant by inequality, because olympiad inequalities don't look any more viable than brute force.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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