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  1. #26
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    Re: HSC 2018 MX2 Integration Marathon

    only requires u substitution.
    Last edited by mrbunton; 2 Apr 2018 at 12:08 PM.

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    Re: HSC 2018 MX2 Integration Marathon





    Justifying this substitution:

    has a solution for only when and .

    The domain of the integrand is and .

    does not exist only when . But since this range is undefined in the integrand to begin with, the substitution is valid.

    So,



    Substituting, we get



    Not sure if that second last step was valid though.

    (edit: oops forgot to sub back in)

    And so



    and by considering the appropriate right angled triangle, . Since that's always positive, the sign function on the left can be eliminated.

    So we have

    Last edited by fan96; 3 Apr 2018 at 6:54 PM.
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    Re: HSC 2018 MX2 Integration Marathon

    an easier way is to just use the substitution u = sqrt(x)
    the integrand will be 2/sqrt(u^2+1), which is trivial to integrate (those usual log ones)
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    Re: HSC 2018 MX2 Integration Marathon

    correct but the two tan signs are generally said to cancel each other out; although i understand it can vary from +-1. that being said if u look at example at the integral tanx/|tanx| it is a horizontal line being split up into positive and negative in pi/2 segments at a time and hence u should not be allowed to take it out theoretically (although area stays the same; definite integral get affected from my understanding)but discontinuous graphs in integration are probably beyond the scope of even ext2(when dealing with indefinite integrals btw; or i havent been taught it yet). Its easier just to cancel the tan out imo at this level.

    im not completely sure but in the solution i got had the tan cancelled out
    Last edited by mrbunton; 4 Apr 2018 at 9:30 PM.

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    Re: HSC 2018 MX2 Integration Marathon

    is anyone else seeing a bunch of extra addition symbols lodged between every character or is that just me
    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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    HSC 2018 MX2 Integration Marathon

    Quote Originally Posted by jathu123 View Post
    an easier way is to just use the substitution u = sqrt(x)
    the integrand will be 2/sqrt(u^2+1), which is trivial to integrate (those usual log ones)
    may as well just jump ahead and multiply by
    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

    Quote Originally Posted by Paradoxica View Post
    is anyone else seeing a bunch of extra addition symbols lodged between every character or is that just me
    Yeah I'm seeing that and it's really confusing


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    Re: HSC 2018 MX2 Integration Marathon

    How to do this question https://imgur.com/a/jrRra

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    Re: HSC 2018 MX2 Integration Marathon

    is it 13/15 pi u^2 ??

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    Re: HSC 2018 MX2 Integration Marathon

    Quote Originally Posted by CapitalSwine View Post
    How to do this question https://imgur.com/a/jrRra
    We can get the volume of the solid by taking the volume of from 0 to 1 (rotated about the y-axis) and then subtracting the volume of the cylinder inside, which has radius and height 1.

    Last edited by fan96; 12 Apr 2018 at 7:44 PM.
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    Re: HSC 2018 MX2 Integration Marathon

    Can someone give me a hint for this question?

    Last edited by fan96; 7 May 2018 at 7:36 PM.
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    Re: HSC 2018 MX2 Integration Marathon

    These are nice practice:



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    Re: HSC 2018 MX2 Integration Marathon

    Quote Originally Posted by fan96 View Post
    These are nice practice:







    The second one is done similarly but with sub instead of IBP
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    Re: HSC 2018 MX2 Integration Marathon

    2017
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    Course: AdvSci(Hons)/Engineering(Hons) @ UNSW

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    Re: HSC 2018 MX2 Integration Marathon

    Quote Originally Posted by jathu123 View Post






    Note that sinh(-x) = 2^-1.(exp(-x) - exp(x)) = -2^-1.(exp(x)-exp(-x)) = -sinh(x)
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    Re: HSC 2018 MX2 Integration Marathon


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    Re: HSC 2018 MX2 Integration Marathon

    Quote Originally Posted by mrbunton View Post
    Are you sure the integrand is correct?

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    Re: HSC 2018 MX2 Integration Marathon

    directly from my test; i couldn't solve it nor do i know the answer however, despite me spending over a hour on this problem.

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    Re: HSC 2018 MX2 Integration Marathon

    Quote Originally Posted by mrbunton View Post
    Surely it's been written wrong? Only way I can think of using Taylor, but that is outside 4U scope.

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    Re: HSC 2018 MX2 Integration Marathon

    Lets campaign the teachers to give mrbunton the mark!

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    Re: HSC 2018 MX2 Integration Marathon

    In the case of a misprinted question, I think the best thing to do here would just be to give everyone the marks.

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    Re: HSC 2018 MX2 Integration Marathon

    Quote Originally Posted by si2136 View Post
    Surely it's been written wrong? Only way I can think of using Taylor, but that is outside 4U scope.
    Try replacing both e^x with e^(x^2).
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    Re: HSC 2018 MX2 Integration Marathon

    Quote Originally Posted by mrbunton View Post
    the question was misprinted on the exam paper, everyone got a mark. the correct version had a x at the bottom instead of a e^x which is easy to solve. i noticed u could find the value if it's in the form of im pretty sure

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    Re: HSC 2018 MX2 Integration Marathon

    The above question is literally impossible, in the sense there is no elementary closed form in terms of a finite combination of elementary operations and HSC-level constants.
    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

    Quote Originally Posted by mrbunton View Post
    Quote Originally Posted by Paradoxica View Post
    The above question is literally impossible, in the sense there is no elementary closed form in terms of a finite combination of elementary operations and HSC-level constants.


    sub into the latter u=-x; dx/du=-1 and limits:

    if i didn't make a mistake(i apologise if i did); the answer should be one step away.
    Last edited by mrbunton; 13 Sep 2018 at 10:35 AM.
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