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Re: HSC 2013 3U Marathon Thread And these sort of expressions are usually integrated by cases.

Re: HSC 2013 3U Marathon Thread The insides being trigonometric has nothing to do with it, it is just that \sqrt{u^2}=|u|.

Re: HSC 2013 3U Marathon Thread Both wolfram alpha and realisenothing aren't entirely correct. This sort of integral is really awkward to find an indefinite integral valid over the whole real line (if it is even possible). Actually I suppose it probably is, using floor/ceiling functions, but...

Re: HSC 2013 3U Marathon Thread and then? ;).

Re: HSC 2013 4U Marathon no, its not.

Re: MX2 Integration Marathon Haha started my post before I saw yours.

Re: MX2 Integration Marathon f(x)=\frac{\pi^2}{8\tan^{-1}(\pi/2)} Or did you want any additional conditions on this function?

Re: HSC 2013 4U Marathon A bijection (wikipedia the term). The problem here is that as x ranges from -1 to 1, u hits every value in (0,1] TWICE instead of once. You resolved this by splitting the integral into two intervals on which u doesn't have this behaviour. (Also, naively looking at...

Re: HSC 2013 4U Marathon Essentially. The main point is that in the changes of variables theorem you need a 1-1 correspondence between your old variable and your new variable...a point often glossed over by teachers.

Re: HSC 2013 4U Marathon Good :).

Re: HSC 2013 4U Marathon Prove that odd functions have even derivatives and vice versa. Is it possible for a function that is NOT odd to have an even derivative?

Re: HSC 2013 4U Marathon True or false: a function that is differentiable over the interval [a,b] is integrable over the interval [a,b].

Re: HSC 2013 4U Marathon Find the error (if any) in the following working: I:=\int_{-1}^1 x^2e^{-x^2}\, dx \\ \\ $Let $u=x^2.\\ \\ \Rightarrow I=\int_1^1 \frac{xe^{-u}\, du}{2}= \int_1^1 \frac{e^{-u}\sqrt{u}}{2}\, du\\ \\ $as $u>0$ in our interval of integration. Hence $I$, having been...

Nah because I may or may not have obtained these games through questionable means. The built in online multiplayer, or LAN using something like Hamachi if necessary.

Anyone up for some multiplayer AoE2? I am horrible at the game but its still pretty fun, especially in big games with teams. (I use the original game plus this expansion http://www.forgottenempires.net/ which contains a few more civs and other features and adds HD support, rather than the...

Re: HSC 2013 4U Marathon LHS=nx_i^2=RHS :P.

This is an obvious disadvantage of using that sign for division rather than \frac{6}{2(1+2)}. I don't think either answer is universally correct here as I have heard order of operations being taught as BOMDAS as well. When you want to be clear, use excessive brackets...it is your job as a...

Re: HSC 2013 4U Marathon (Sorry for being so lazy, but I guess this thread is meant for current students. Hopefully my sketch will help someone actually doing their HSC to do the question!) If no-one uploads an actual answer I'll upload my handwritten solution later.

Re: HSC 2013 4U Marathon I will provide a sketch of a proof. I encourage anyone who hasn't done this question to still try it though, it is good even when viewed as a purely algebraic exercise. 1. This is a standard integration by parts induction. We use I_n to denote this integral from...

Re: HSC 2013 4U Marathon Tricky question! Am clearly rusty at contest mathematics. $By a straightforward induction we get $f(g_n(x))=f(x)$ for all $x$, where $g_n(x)=x+nf(x)$. Now we claim that each $g_n$ is injective.\\ $g_n(x)=g_n(y)\Rightarrow f(x)=f(g_n(x))=f(g_n(y))=f(y)\Rightarrow...