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Thread: HSC 2018 MX2 Integration Marathon

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    Re: HSC 2018 MX2 Integration Marathon









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    Re: HSC 2018 MX2 Integration Marathon

    Show that


    Last edited by stupid_girl; 28 Oct 2018 at 10:28 PM.

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    Re: HSC 2018 MX2 Integration Marathon



    Using the identities:





    the integral reduces to:





    Perform a substitution to get



    The substitution would probably have given a quicker answer, but the first thing that came to mind was another trig sub:



    Giving:



    Noting that for all real and so for the bounds of this integral, we may simplify to get





    Note: and can be evaluated by using an appropriate right angled triangle.





    Expanding the square and rationalising the denominator gives

    Last edited by fan96; 11 Nov 2018 at 12:45 PM.
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    Re: HSC 2018 MX2 Integration Marathon

    Continuing on from the simplification



    Reverse the chain rule twice to obtain:



    Complete the substitution to obtain:



    Using the former table of standard integrals, the integral evaluates to:



    Which "simplifies" to:

    Last edited by Paradoxica; 11 Nov 2018 at 3:26 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.

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    Re: HSC 2018 MX2 Integration Marathon

    Maybe something a bit easier:

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    Re: HSC 2018 MX2 Integration Marathon

    Damn that pi lemme sit on it a n=bit longer

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    Re: HSC 2018 MX2 Integration Marathon

    Show that
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    Re: HSC 2018 MX2 Integration Marathon

    Quote Originally Posted by HeroWise View Post
    Damn that pi lemme sit on it a n=bit longer
    Just consider the integral from 0 to a first, then take the limit as a->pi-.

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    Re: HSC 2018 MX2 Integration Marathon

    Quote Originally Posted by stupid_girl View Post
    Show that
    If you don't mind me asking, where do you get these integrals from?
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    Re: HSC 2018 MX2 Integration Marathon

    I put my solution as an attachment so that it won't spoil the answer for other people attempting to solve this.
    Attached Files Attached Files
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    Re: HSC 2018 MX2 Integration Marathon

    Quote Originally Posted by fan96 View Post
    If you don't mind me asking, where do you get these integrals from?
    Construct from simpler integrals using trig identities and properties of definite integrals.
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    Re: HSC 2018 MX2 Integration Marathon

    Continue to have fun with trig.

    Harder version:

    Find the area between x-axis and y=f(x) on its maximal domain.

    Simpler version:
    Show that
    Last edited by stupid_girl; 16 Dec 2018 at 10:32 PM.

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    Re: HSC 2018 MX2 Integration Marathon

    Quote Originally Posted by stupid_girl View Post
    Continue to have fun with trig.

    Harder version:

    Find the area between x-axis and y=f(x) on its maximal domain.

    Simpler version:
    Show that
    a very nice integral.
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    Re: HSC 2018 MX2 Integration Marathon

    This one should be considerably easier than the previous one.



    The answer is pretty small. (1/32304)
    Last edited by stupid_girl; 20 Dec 2018 at 7:31 PM.

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    Re: HSC 2018 MX2 Integration Marathon

    I've reduced the integral to



    if someone else wants to finish it from this, but it seems very difficult.

    Maybe a different approach might be necessary?
    Last edited by fan96; 23 Dec 2018 at 2:01 PM.
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    Re: HSC 2018 MX2 Integration Marathon

    Quote Originally Posted by fan96 View Post
    I've reduced the integral to



    if someone else wants to finish it from this, but it seems very difficult.

    Maybe a different approach might be necessary?
    You are almost there. If you put x=tan theta, does it look familiar? You've solved that in the previous one.
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    Re: HSC 2018 MX2 Integration Marathon





    By using



    and some manipulation, similar to the previous integral, we get









    Because the integrand is even,







    Integrating by parts,







    Because for ,



    This integral has been evaluated before to be equal to .

    Last edited by fan96; 24 Dec 2018 at 3:16 PM.
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    Re: HSC 2018 MX2 Integration Marathon

    This one is absolutely a beast.

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    Re: HSC 2018 MX2 Integration Marathon

    I have a question, When do you know to use
    Last edited by HeroWise; 12 Jan 2019 at 5:41 PM.

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    Re: HSC 2018 MX2 Integration Marathon

    Quote Originally Posted by HeroWise View Post
    I have a question, When do you know to use
    when f(x)+f(a-x) is easier to integrate than f(x)

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    Re: HSC 2018 MX2 Integration Marathon

    This is another beast.

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