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Re: MX2 Integration Marathon u^2 = \frac{x}{x+a} x = -a\frac{u^2}{u^2-1} dx = \frac{2a u}{(u^2-1)^2} \ du I = 2a \int \frac{u \sin^{-1}u}{(u^2-1)^2} \ du u = \sin \theta I = 2a\int \frac{\theta \sin \theta}{\cos^3 \theta} \ d\theta Integrating by parts u = \theta \ \ dv = \tan...

Re: MX2 Integration Marathon You have done it for when there is no square root inside the asine function. =============== \int_0^{\infty} \frac{1}{(x^2+x+1)^3} \ dx Remember if you can't be bothered to write a solution, a summary at each major step is sufficient. (i.e. Do IBP to get...

Re: HSC 2013 4U Marathon You could do that, but of course there are ways of proving that result without using Binomial theorem (i.e. combinatorics), iirc one of your questions on here was about that.

Re: HSC 2013 4U Marathon $i) Prove that$ \ \ \int_{0}^{\frac{\pi}{2}} \sin^{2n-1} x \ dx = \frac{2 \cdot 4 \cdot 6 \dots (2n-2)}{1 \cdot 3 \cdot 5 \dots (2n-1)} \ \ \ \fbox{2} $ii) Hence or otherwise prove that$ \sum_{n=1}^{\infty} \frac{1}{\binom{2n}{n}} = \frac{1}{3} + \frac{2\pi...

Re: HSC 2013 4U Marathon Oops heh, fixed.

Re: HSC 2013 4U Marathon $Prove without using the Binomial Theorem$ \binom{2n}{n} < 4^n

Re: HSC 2013 4U Marathon For the UNSW question, I have been able to simplify the problem into proving: \int_k^{2k} f(x) \ dx = k^2 k=2012 If this can be done then I have a nice proof in mind.

\int \frac{x^3}{2x+1} \ dx By doing long division, we can see that: \int \frac{x^3}{2x+1} \ dx = \int \frac{(2x+1)(\frac{x^2}{2} - \frac{x}{4} + \frac{1}{8}) - \frac{1}{8}}{2x+1} \ dx = \int \frac{x^2}{2} - \frac{x}{4} + \frac{1}{8} - \frac{1}{8} \cdot \frac{1}{2x+1} \ dx =...

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

Re: HSC 2013 4U Marathon Yep its wrong. Since from the beginning f(x) = c then f(x) = f(x+f(x)) = f(x+c) Since the inside is linear, f'(x) = f'(x+c)

Re: HSC 2013 4U Marathon Because from the very beginning f(x) = 2012, and f'(x) = 0 so f(x+f(x)) = f(x+0)=f(x).

Re: HSC 2013 4U Marathon I'm pretty sure that step is legitimate. We are doing the same operation to both sides, if f(x) = g(x), then f'(x) = g'(x)? or am I missing something? EDIT: wtf man what am I thinking Oh and Realise, nice work for my question I had the same method.

Re: HSC 2013 4U Marathon Yep those are the only solutions, what was your method?

Re: HSC 2013 4U Marathon $For some number$ \ \ a f(a) = f(a+f(a+f(a+f(a+ \dots +f(a)) \dots ) = f(k) for some number k, f is continuous and defined on all real therefore a and k can be anything. If that is the case then a and k can be such that they represent any pair of real numbers. The...

Re: HSC 2013 4U Marathon $By DeMoivere's theorem$ (\cos x + i\sin x)^k = \cos(kx) + i\sin(kx) (1+ i\tan x)^k = \frac{\cos(kx)}{\cos^kx} + i \frac{\sin(kx)}{\cos^k x} $Im$ \left(\sum_{k=1}^{n} (1+i\tan x)^k \right) = \sum_{k=1}^{n} \frac{\sin kx}{\cos^kx} \ \ \dots \ \ (1) $Consider the...

Re: MX2 Integration Marathon I just got it off a random site, as a proof that 22/7 exceeds pi.

Re: MX2 Integration Marathon \int_0^1 \ln \left(\sqrt{1-x} + \sqrt{1+x} \right ) \ dx

Yeah that's true. But the difference here is that incomplete answers are given very little credit, whereas in the HSC one can just pick out all the easy marks from a hard question.

Too many free marks, way too much hand-holding and spoon-feeding.

Did you look at: http://www.admissionstestingservice.org/our-services/subject-specific/step/preparing-for-step File sizes are quite small. Also with regards to the content in STEP papers (someone please correct me if I'm wrong) but the Mechanics and some statistics is out of the syllabus for...