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Re: MX2 2016 Integration Marathon $\textbf{ALTERNATE METHOD}$ $Let $ \mathcal{I} = \int x^2 \cot ^{-1} x \text{ d}x$. Using IBP by integrating the $x^2$ and differentiating the $\cot ^{-1} x$ yields$ $$\begin{align*} \mathcal{I} &= \frac{x^3}{3}\cot ^{-1}x -\frac{1}{3} \int x^3 \cdot...

Re: HSC 2016 4U Marathon $This is because an angle at the circumference of a circle must be between 0 and $\pi$. So if the absolute value of the R.H.S. given to you is not in this range, we need to \textsl{add a multiple of} $2\pi$ to it to get it into the range $(0,\pi)$.$ $The reason we can...

$If $x\neq 0$, we can just simplify it (if $x=0$, the expression is undefined). For example, we can simplify something like $\frac{x^3}{x}$ to $x^2$ if $x\neq 0$ (basically index laws; `cancel' an $x$ from the numerator and denominator to be left with $\frac{x^2}{1}$, which is $x^2$). A similar...

$For $x\neq 0$, we can simplify it to $9x^2 + x+1$, which is a polynomial in $x$.$

Re: MX2 2016 Integration Marathon $Note that the integrand given in Ekman's question can be simplified to $\frac{1+x^2}{2+x^2}$, which you can show using right-angled triangles.$

Surely not necessary for 30 out of 50 in HSC 3U Maths? A 30 out of 50 is like a Band 3. I'm not sure what it is in terms of E's for 1 unit subjects (probably E2 to E3?).

Re: MX2 2016 Integration Marathon Alternatively, we can again do it by breaking up the integral and using IBP to cancel out some integrals and be left with the answer. This would require a tedious differentiation using the quotient rule, but avoids the need for non-well-known trig. identities.

Re: HSC 2016 4U Marathon $Note however that if $z = r\, \mathrm{cis} (\theta)$ is a general non-zero complex number and $\alpha = \frac{1}{n}$ for some positive integer $n$ (where $r,\theta \in \mathbb{R}$), then $z^{\alpha}$ will be \textsl{multivalued}. This means that we need to assign a...

Re: HSC 2016 4U Marathon $Use De Moivre's Theorem. Note that in general powers of $z$ like this (i.e. non-integer powers) are multivalued and the idea of principal values of these is not in the syllabus as far as I'm aware. So $\frac{1}{2} \arg (z)$ would be the argument of one of the square...

Re: MX2 2016 Integration Marathon Well my method doesn't require those trig. identities like in your second method, but the idea is similar (cancel out one integral by using the integral from the IBP step).

Re: MX2 2016 Integration Marathon $Essentially, here is a way to do it without just completely guessing the answer.$ $Write $\mathcal{I} = \int \frac{1-\sin x }{1-\cos x}\cdot \mathrm{e}^x \text{ d}x$ as $\mathcal{I} = \int \frac{\mathrm{e}^x}{1-\cos x}\text{ d}x - \int \frac{\sin x}{1- \cos...

Re: HSC 2016 3U Marathon $\underline{NEW QUESTION}$ $(i) Show via mathematical induction that $\cos \left(n\pi\right) = (-1)^n$ for all non-negative integers $n$.$ $(ii) Deduce that the result in (i) holds for all integers $n$.$

Lol, inb4 Ekman or Gabriel Moussa references instead. :P

Re: HSC 2016 4U Marathon $\textbf{NEW QUESTION}$ $Sketch (or for this Marathon, describe) the locus given by $\arg \left(z^3\right) = \frac{\pi}{3}$.$

Re: HSC 2016 4U Marathon $\textbf{NEW QUESTION}$ $If $z$ is a complex number and $z\neq 0$, what does `$\arg (z)$' refer to? Does $\arg (z) = \arg (z) + 2\pi$? If so, does $0=2\pi$? Explain what is going on here.$

Re: HSC 2016 4U Marathon $The rule for sketching the locus of $\arg \left(\frac{z -z_1}{z -z_2}\right)= \alpha$ for a given $\alpha \in \left(0,\pi\right)$ is to start at the point $z_1$ and go counter-clockwise in a circular arc to the point $z_2$, with an angle of $\alpha$ being subtended in...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $The rule for $\arg \left(\frac{z -z_1}{z -z_2}\right)=\alpha$ for $\alpha \in \left(0,\pi\right)$ is to start at the point $z_1$ and go counter-clockwise in a circular arc to the point $z_2$, with an angle of $\alpha$ being...

Re: MX2 2016 Integration Marathon $Correct!$ $\textbf{NEW QUESTION}$ $Find $\int \frac{9 +6\sqrt{x} + x}{4\sqrt{x} + x}\text{ d}x$.$

Re: MX2 2016 Integration Marathon $\textbf{NEW QUESTION}$ $Define $f_1 (x) = |x|$ and for integers $n\geq 2$, $f_n (x) = \Big{|} x + f_{n-1} (x)\Big{|}$.$ $Let $\mathcal{I}_n =\int _{-1} ^{1} f_n (x) \text{ d}x$, for $n= 1,2,3,\ldots$.$ $Derive an expression for $\mathcal{I}_n$ in...

Re: MX2 2016 Integration Marathon $\underline{NEW QUESTION}$ $Find $\int \sqrt{x + \sqrt{x^2 + 1}} \text{ d}x$.$