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Re: HSC 2017 MX2 Integration Marathon $\noindent Apparently they mark incredibly harshly in the Putnam. Could that have contributed to it? Would they have perhaps given low scores even to people who obtained the correct answer of $\frac{\pi \ln 2}{8}$ (maybe because they skipped some steps...

Re: HSC 2017 MX2 Integration Marathon $\noindent Those two tan expressions aren't equivalent, but their integrals from 0 to $\frac{\pi}{4}$ are equal.$ $\noindent Geometrically speaking, integrating $f(x)$ from $x = 0$ to $a$ yields the same result as integrating $f(a- x)$ from $x = 0$ to $a$...

Re: Extracurricular Integration Marathon This paper may be of interest: https://cs.uwaterloo.ca/journals/JIS/VOL17/Neto/neto4.pdf .

Re: HSC 2017 MX2 Integration Marathon $\noindent It's the rule $\int \frac{u'v -uv'}{v^{2}} \,\mathrm{d}x =\frac{u}{v} + c$.$

Re: HSC 2017 MX2 Integration Marathon Why was it so poorly done in the Putnam? I thought the integral would have been relatively famous (and not that difficult either). Or were a lot of people scared off by it being A5?

You may want to keep in mind the three variable result a^{3} + b^{3} + c^{3} - 3abc = \left(a + b + c\right)\left(a^{2} + b^{2} + c^{2} - ab - bc - ca\right). \noindent \text{This can also be used to prove three-variable AM-GM, by first showing} \ (a^{2} + b^{2} + c^{2} \geq ab + bc + ca \...

Yes, there is. In fact, there is a recursive formula for all sums of non-negative integer powers (called \textbf{power sums}) in terms of lower powers' sums and elementary symmetric polynomials (expressions like sum of roots, sum of pairs of roots, etc., which are easily computable given a...

Re: HSC 2017 MX2 Integration Marathon It's just reverse chain rule (use a substitution u = cos(2x) if in doubt).

Re: HSC 2017 MX2 Integration Marathon Since the denominator has a cos-squared, the numerator isn't going to be like the derivative of the denominator (it would've been though if the denominator just had cos without the square). Instead, we have a function that is (proportional to) u'/(1...

Re: HSC 2017 MX2 Integration Marathon $\noindent Find $\int \sqrt{1 + e^{x}}\, \mathrm{d}x$.$

Re: HSC 2017 MX2 Integration Marathon $\noindent Find $\int \frac{\cos x + \sin x}{3\cos x + 4\sin x}\, \mathrm{d}x$.$

Re: Several Variable Calculus Sorry misread it (saw + instead of -). One way is to recall that the preimage of an open set under a continuous map is open.

Re: Several Variable Calculus That set Omega is an open ball already (using the Euclidean metric), and an open ball is an open set (you should prove this as an exercise if you haven't before).

\noindent \text{It's worth knowing that} \ a^2 + b^2 + c^2 + d^2 = \left(a+b+c+d \right)^{2} -2\left(ab + bc + cd + da + ac + bd \right). \text{In general,} \begin{align*}\sum_{i = 1}^{n} a_{i}^{2} &= \left(\sum_{i=1}^{n} a_{i}\right)^{2}-2\sum_{1\leq i < j\leq n}a_{i}a_{j}.\end{align*}...

Re: Several Variable Calculus Some rational points arbitrarily close to sqrt(2) are points where we truncate the decimal expansion of sqrt(2) arbitrarily far. I.e. $$\lfloor 10^{n}\sqrt{2}\rfloor 10^{-n}$$ for positive integers n.

Music involves superposition of waves. (Finding the component frequencies making up a signal leads into something called Fourier Analysis. You can read more about these things here: http://cmc.music.columbia.edu/musicandcomputers/chapter3/03_03.php .)

Re: Several Variable Calculus $\noindent Here's the gist of a way to do it. For $x\neq 0$ and $y\neq 0$, we have$ $$\begin{align*}\left|\frac{\sin (xy)}{x} - a \right| &= \left|y\right|\left|\left(\frac{\sin (xy)}{xy} -1\right) + \left(1 - \frac{a}{y}\right) \right| \\ &\leq \left|y \right|...

$\noindent The equation is equivalent to $\log_{x} \left(7^3\right) = \ln 10$ (using the log law $c\log a = \log a^{c}$ on the left-hand side). This means that $7^{3} = x^{\ln 10}$. Thus the solution is $\boxed{x = \left(7^{3}\right)^{\frac{1}{\ln 10}} = 7^{\frac{3}{\ln 10}}}$.$

$\noindent Make use of the property $\left(\mathbf{u}^{T}M\mathbf{v}\right)^{T} = \mathbf{v}^{T}M^{T}\mathbf{u}$.$

Here's some hints. $\noindent a) Show $\mathbf{x}^{T}G^{T}\mathbf{x} =\mathbf{x}^{T}G\mathbf{x}$ for all $\mathbf{x}\in \mathbb{R}^{n}$ (where $n$ is the matrix size), e.g. by considering properties of transpose / inner products or whatever method you like.$ $\noindent b) One way is to...