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Re: MX2 Integration Marathon Well yeah if its right its right, nice work. Alternatively the substitution u=1+x^3 and the split of the first x^5 into x^3 and x^2 yields a faster method.

Re: MX2 Integration Marathon \int_0^1 x^5 \sqrt{1+x^3} \ dx

Re: MX2 Integration Marathon Yep, I was thinking of (u-1)^2 which is slightly longer. =========== \int \tan^{-1} \left(\frac{1}{1+x(x-1)} \right ) \ dx

Re: MX2 Integration Marathon The substitution (u^2-1)^2 can be modified slightly to make the integral much more easier. Its close though.

Re: HSC 2013 4U Marathon You are missing EE :P

Re: HSC 2013 4U Marathon The amount of words starting with A, B, C, D is: 4 \cdot 26^2 The amount of words starting with E, middle of A, B, C, D, E or F is 26 \cdot 6 The amount of words starting with E, middle of G before EGG is 6. So final answer: \fbox{2866} ======...

Re: HSC 2013 2U Marathon Made it easier.

Re: MX2 Integration Marathon Well my intent for the question was for someone to say f(x) = atan(x), but I guess I just posed this without really thinking about it :s

Re: MX2 Integration Marathon lol nope hehe A harder one: \int \sqrt{1+\sqrt{x}} \ \ dx

Re: MX2 Integration Marathon I = \int_0^{\pi /2} \frac{f(x) }{1+x^2} \ dx = \frac{\pi^2}{8} $Find an example of a function$ \ \ f(x) \ \ $ such that the integral holds$

Re: HSC 2013 3U Marathon Thread That isn't correct.

Re: HSC 2013 3U Marathon Thread Lol man its called a mistake, I thought the way the conditions were layed out a team could not be constructed in some cases. Nevertheless I still had a go and it might be wrong but at least I tried.

Re: HSC 2013 3U Marathon Thread There is a fault in your question. And I only just got to look at this question now. edit: I wasn't thinking properly and made a mistake, I was confused with the conditions and I thought that the team couldn't play in some conditions but I see you resolved...

Re: MX2 Integration Marathon \int \sin(\ln x) + \cos(\ln x) \ dx

Re: HSC 2013 4U Marathon Ok yep I get it.

Re: HSC 2013 4U Marathon Can you explain this what you mean by 1-1 correspondence? I haven't heard of this rule with substitution.

Re: HSC 2013 4U Marathon Indeed any odd function plus a (non zero) constant is an example of a non-odd function with an even derivative.

Re: HSC 2013 4U Marathon $Upon the substitution$ u=x^2 $The positive square root is taken of both sides (as evident in the working)$ \sqrt{u} = |x| \sqrt{u} = \begin{cases} x \ \ x\geq 0 \\ -x \ \ x<0 \end{cases} $By definition$ $Hence the proper manipulation would be$ I =...

y = x^3 - 4x y'=3x^2 - 4 x=1 y'= 3-4 = -1 $Since$ \ \ y' \ \ $is the gradient function of the original curve$ \ \ -1 \ \ $is the gradient of the tangent at the point (1,-3)$ $Since by definition, a normal is perpendicular to the tangent, we have$ m_N \cdot m_T = - 1 m_N \ = \...

Re: HSC 2013 2U Marathon Definitely.