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Re: HSC 2013 4U Marathon First lets model the particle travelling in circular motion with parametric equations: x=r\cos \theta y = r\sin \theta Where r is the radius, and theta is the angle the particle makes with the x-axis at some time t. And x and y are the x and y displacements with...

Re: MX2 Integration Marathon \int \frac{(\sqrt{x}+3)^2}{\sqrt{x}(\sqrt{x}+4)} \ dx u= \sqrt{x} + 4 (u-4)^2 = x dx = 2(u-4) \ du \int \frac{(u-1)^2 \cdot 2(u-4)}{(u-4)u} \ du 2\int u - 2 + \frac{1}{u} \ du =u^2 - 4u + 2\ln u + c = (\sqrt{x}+4)^2 - 4(\sqrt{x}+4) +2\ln(\sqrt{x}+4) + c

Re: MX2 Integration Marathon At first I did IBP with u=x^3, dv= x/sqrt(1+x^2) Then did it again, similar length.

Re: HSC 2013 4U Marathon Here is a mechanics one: $A particle is travelling in circular motion, \\ show that its acceleration is given by$ a = r\sqrt{\phi^2 + \omega^4} $Where$ \ \ \omega \ \ $is the angular velocity$ \phi \ \ $is the angular acceleration$ r \ \ $is the radius of motion$

Re: HSC 2013 4U Marathon I think this counts under Polynomials:

Re: MX2 Integration Marathon Yep, nice work. \int_0^{\pi/2} \frac{\sin(2013x)}{\sin x} \ dx EDIT: Forgot limits.

Re: HSC 2013 4U Marathon Just some inequalities, they are not in order of difficulty, nor will the answer to one necessarily help with the others: $Prove that$ $a)$ \ \ (a+b)(a+c)(b+c) \geq 8abc $b)$ \ \ a^4+b^4+c^4 \geq abc(a+b+c) $c)$ \ \ \frac{a+b+c+d}{4} \geq \sqrt[4]{abcd}

Re: MX2 Integration Marathon Is there a USYD Integration Bee? \int \sin(2x) \sin(\cos x) \ dx

Re: MX2 Integration Marathon Yep nice work. 1. Expand denominator: \int \frac{dx}{\frac{1}{2}\sin x \cos x- \frac{\sqrt{3}}{2}\cos^2x} 2. Change to sin2x and cos2x \int \frac{4 \dx}{\sin(2x) -\sqrt{3}(\cos 2x + 1)} 3. Apply the substitution: t=\tan x \int...

Re: HSC 2013 4U Marathon Oops yeah hehe. Coming from x_k/x_(k+1) = x_(k+1)/x_k, and x_k is positive. Hence x_k=x_(k+1)

Re: HSC 2013 4U Marathon LHS - RHS, make it a common denominator, the result simplifies into proving: (x_1+x_2+ \dots + x_n) (\frac{1}{x_1} + \frac{1}{x_2} + \dots + \frac{1}{x_n}) \geq n^2 This can be done by expanding the LHS, you will get, an n number of 1's And \binom{n}{2} pairs of...

Re: HSC 2013 4U Marathon Similar to what I had in mind: \prod \frac{k^2-1}{k^2} = \prod \frac{k-1}{k} \prod \frac{k+1}{k} we are allowed to do this since order in multiplication does not matter. The seperate products all cancel out if you expand it out to yield n+1/2n > 1/2

Re: HSC 2013 4U Marathon Quite clever, I'm unsure about how to prove that: 1-\frac{1}{x^2} > \frac{1}{2^{2^{\frac{1}{j-1}}}} Maybe using calculus, taking f(x) = LHS - RHS, then differentiating etc etc. But differentiating the RHS would be..... yeah lol EDIT: I guess logarithmic...

Re: HSC 2013 4U Marathon Yep. If I want an inductive solution, I will specify that in the question :)

Re: HSC 2013 4U Marathon $Prove that$ \left(1- \frac{1}{4} \right) \left(1- \frac{1}{9} \right ) \left( 1 - \frac{1}{16} \right ) \dots \left( 1 - \frac{1}{n^2} \right ) > \frac{1}{2} \ \ \ \ \fbox{2} (added in mark count as an estimated length of soln)

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

Re: HSC 2013 4U Marathon i) All variables within S_k is positive, all operations are positive, hence S_k > 0 S_k = \frac{a}{k} \left(\frac{1}{n+ b/k} + \frac{1}{n+ 2b/k} + \frac{1}{3b/k} + \dots+ \frac{1}{n+ nb/k} \right) < \frac{a}{k} \left(\frac{1}{n} + \frac{1}{n} + \dots + \frac{1}{n}...

Re: MX2 Integration Marathon Yep nice work.

Economics

Re: MX2 Integration Marathon \lim_{t \to \infty}\int_0^{t} \frac{x}{x^4+1} \ dx