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Re: Extracurricular Integration Marathon $\noindent In order to evaluate this integral we will convert it into a double integral. Observing that$ \int^b_a \sin (ux) \, du = \frac{\cos (ax) - \cos (bx)}{x}, $ the integral can be rewritten as$...

Re: Extracurricular Integration Marathon $\noindent In our solution we will make use of the following two Maclaurin series expansions \tan^{-1} t = \sum^\infty_{n = 0} \frac{(-1)^n t^{2n + 1}}{2n + 1}, \, |t| < 1 $ and $\,\sinh^{-1} t = \sum^\infty_{n = 0} \frac{(-1)^n (2n)! t^{2n +...

Re: HSC 2017 MX2 Integration Marathon $\noindent Ask yourself the question what is the integral $\int \frac{dt}{t}? $\noindent Of course you will immediately say it is the natural logarithm! Well this is just a convenient way of papering over the fact that until we expanded our function...

Re: HSC 2017 MX2 Integration Marathon $\noindent When we say an integral cannot be integrated in \textit{elementary terms} it means no primitive for the function can be found in terms of the so-called elementary functions. The elementary functions are polynomials, roots, exponential...

$\noindent \textbf{Proof for the second part}$ $\noindent Squaring the equation $|z - z_1| = k|z - z_2|$ and making use of the result $w \bar{w} = |w|^2$ where $w \in \mathbb{C}$ we have$ \begin{align*}|z - z_1|^2 &= k^2|z - z_2|^2\\(z - z_1)\overline{(z - z_1)} &= k^2(z - z_2)\overline{(z...

$\noindent The general equation of a circle in the complex plane should be $ \alpha z \bar{z} + \beta z + \bar{\beta} \bar{z} + \gamma = 0. $\noindent \textbf{Proof for first part}$ $\noindent As should be well-known, the complex numbers $z$ satisfied by the set of points given by$...

$\noindent Recall a continuous function $f$ is said to be increasing if $f'(x) > 0$ (namely, it derivative is positive) for all values of $x$ on some interval. $\noindent Now let $f(x) = \sin x - x + \frac{x^3}{4}$. So$ \begin{align*}f'(x) &= \cos x - 1 + \frac{3x^2}{4}\\f''(x) &= -\sin x +...

Re: HSC 2017 MX2 Integration Marathon $\noindent While you may not know a particular method explicitly, it is possible in a Question 16 type of question in the HSC you could be \textit{guided} through using a previously unknown technique or method.

Re: HSC 2017 MX2 Integration Marathon $\noindent You might like to try out the rule on the following question. And just for fun, let us find this integral in three different ways with a few hints provided along the way.$ $\noindent Find $\int \frac{x^2}{(x \sin x + \cos x)^2} \, dx $ as...

Re: Extracurricular Integration Marathon $\noindent We begin with a few preliminary results that I will make use of in my derivation.$ \int^\infty_0 e^{-x^2} \, dx = \frac{\sqrt{\pi}}{2}$ (the Gaussian integral)$ \int^\infty_0 e^{-2x^2} \, dx = \frac{\sqrt{\pi}}{2\sqrt{2}}$ (Substitute $u =...

Re: International Baccalaureate Marathon $Not sure if you know about the \textit{vector cross product}, but if you do one of the geometric interpretations for the magnitude of the cross product between two vectors corresponds to the area of a triangle.$ $If two sides of a triangle are...

A few slots are still available for anyone else who may be interested.

Are offering tuition in Preliminary/HSC Mathematics (2U), Mathematics Extension 1 (3 Unit), and Mathematics Extension 2 (4 Unit). Are also available for tutoring in First-Year University Mathematics. About Me My name is Sean and I have 15+ years teaching experience at the secondary and...

By definition, all 4 Unit students are 3 Unit students already so would naturally be allowed to choose Statistics in Year 12 if they so wished.

Or perhaps statistics could be added as an additional unit at HSC level only open to students studying either 2 or 3 Unit. In this way you would only be drawing on your existing pool of 2 and 3 Unit students rather than Statistics having to compete against them for students. It would be quite a...

One of the three options being proposed for physics (and chemistry as well) late last year which BOSTES were seeking feedback on, was a return to a standard, introductory-level physics course (essentially something akin to what we had pre-2001). Although the proposal was a little light on...

I am guessing many students find first-year physics harder than 4 Unit Maths since it is most likely their first ever exposure to real physics. There is absolutely no relation between the real physics taught at universities with that light weigh puff piece known as HSC Physics. The latter is...

Re: Help i'm trapped in the title and can only speak once every 2 years $\noindent I will assume $n \neq 0. \begin{align*}I &= \int 20n x^{5n-1} \tan^{-1}\left( \frac{(x^n-1)x^n(x^n+1)}{(x^{2n}-\sqrt{3}x^n+1)(x^{2n}+\sqrt{3}x^n+1)} \right)\,dx\\&= \int 20 n x^{5n - 1} \tan^{-1} \left...

Re: MX2 2016 Integration Marathon $\noindent For those who wish to keep it purely real, various product-to-sum and sum-to-product identities for the trig functions can be used to find each of the sums. For the general case we will consider $\frac{\sum^{n - 1}_{k = 1} \sin k\theta}{\sum^{n -...

Re: MX2 2016 Integration Marathon $\noindent \textbf{Next Question} $\noindent Find $\int \frac{\sin x + \sin 2x + \sin 3x + \sin 4x}{\cos x + \cos 2x + \cos 3x + \cos 4x} \, dx