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Re: HSC 2013 4U Marathon $Prove that$ \sum_{k=0}^{n} \binom{n}{k}\frac{(-1)^{k}}{2k+1} = \frac{ (n!)^2}{(2n+1)!}2^{2n} (-1)^n ii) It is obvious that one simply needs to pick the right function f(x), it is clear then that we need to find some function f so that: \frac{1}{n}f( \left(1+...

Re: MX2 Integration Marathon \int \sqrt{\frac{a-x}{x-b}} \ dx

Re: MX2 Integration Marathon Yeah, relative to most other schools, definitely.

Re: MX2 Integration Marathon 'newer papers are easier' ; ) Seriously though, the tan(x/2) substitution is obvious, and the tan inverse identity is obvious, the only thing even requiring a sort of intuition is the use of double angle formulae while simplifying which in a sense is obvious when...

Re: MX2 Integration Marathon Yes you are correct, apologies.

Re: MX2 Integration Marathon Originally STEP 200(something)! Fun fact, some CSSA questions are directly ripped from a STEP paper. (i.e. 2011 Q7a, 2011 Q8b) That is the only CSSA paper that I've seen so far, I'm going to start doing the other ones soon, but I'm guessing that even from the...

Re: MX2 Integration Marathon $Prove$ \int_0^{\pi /2} \frac{dx}{1 + \cos \alpha \sin x} = \frac{\alpha}{\sin \alpha} \cos \alpha \neq \pm 1

Re: HSC 2013 4U Marathon $For some function$ \ \ f(t) \ \ $the operation$ \ \ L \ \ $is applied, the resultant function is$ F(s) = \lim_{z \to \infty} \int_0^z e^{-st} f(t) \ dt \ \ \ \ \ (s>0) $Operation$ \ \ L \ \ $can only be applied to a function if its resultant function converges to...

Re: MX2 Integration Marathon Yeah that's true....

Re: MX2 Integration Marathon Yep this is what I was aiming at, divide both top and bottom by x^2, then substiute u=x+1/x, it dissolves into a standard integral. Asianese and Realise your answers may be of a different form that can be manipulated using log properties or something.

Re: MX2 Integration Marathon I don't think that is correct, some of your logarithms look similar to my ones so you may have made a mistake somwhere.

Re: MX2 Integration Marathon \int \frac{x^2-1}{x\sqrt{x^4+1}} \ dx

First one: We cannot deal with that series as a whole, so its best to split it into 2 separate ones: S=1-2+3-4+ \dots - 100 = (1+3+5+ \dots +99) - (2 + 4 + \dots + 100) Each of them are arithmetic series each with a common difference of 2 (but different initial term). So using the sum...

bump

http://community.boredofstudies.org/showthread.php?t=290055

Re: HSC 2013 4U Marathon Ah alright then :s Second part is still open though.

Re: HSC 2013 4U Marathon Is there a problem with the argument; that considering finite n, the sum of the lower rectangles of width 1/n is approximately the area (which is the integral), and then saying as n increases without bound the rectangles all sum up to become the area? Similar logic is...

Re: HSC 2013 4U Marathon I don't see a need to but feel free to do so if it will help.

Re: HSC 2013 4U Marathon $i) Prove$ \int_1^2 f(x) \ dx = \lim_{n \to \infty} \left(\frac{1}{n} \sum_{k=1}^{n}f \left(1 + \frac{k}{n} \right ) \right ) $ii) Hence evaluate$ \lim_{n \to \infty} \left(\frac{n}{2n^2 + 2n + 1} + \frac{n}{2n^2 + 4n + 4} + \frac{n}{2n^2 + 6n + 9} + \frac{n}{2n^2...

Re: HSC 2013 3U Marathon Thread Some questions I do genuinely make (not this one). Otherwise I will sometimes steal from STEP/Internet/modified HSC/trials. I got that from a 4U trial Q8 and removed the first part of the question. Which just goes to show how weak of a question it was lol.