Search results

At least quick partial fractions is possible this time round lol u^2 = \tanh x \implies 2u\,du = \text{sech}^2 x\,dx = (1-\tanh^2 x)\,dx\quad (u\ge 0) \begin{align*} I & = \int \sqrt{ \tanh x} \, dx\\ &= \int \frac{2u^2}{1-u^4} \, du\\ &= \int \frac{2u^2}{(1-u^2)(1+u^2)}\,du\\ &=...

Re: HSC 2018 MX2 Integration Marathon \text{For }a>0,\text{ find }\int_0^{\infty} \frac{dx}{(1+x^a)(1+x^2)} Also spoiler for the one above:

Yes.

It's a bit lame but it's not unjustified, because a lot of people rote-learn the (n-1)! formula without understanding why it works. Thus they can't adapt it to other circular arrangements (e.g. those, but two must sit next to each other).

Easy difficulty \int \frac{\tanh x}{\exp x}\,dx

\text{Show that }\int_0^{2\pi} (\cos \theta)^{2n}\,d\theta = \frac{\pi}{2^{2n-1}}\binom{2n}{n} (Try ignoring the lengthy 4U way in this thread :P)

Re: HSC 2018 MX2 Integration Marathon \int_0^\infty \frac{dx}{\left(x+\sqrt{1+x^2}\right)^n}

Re: HSC 2018 MX2 Integration Marathon \text{Suppose that }0\le x \le y \\\text{Since }t\in [0,\pi] \implies \sin t \in [0,1]\\ x\le y \implies \sin^x t \ge \sin^y t \\\text{Hence for }t\in [0,\pi]\\ t\sin^x t \ge t \sin^y t \implies \int_0^\pi t\sin^x t\,dt \ge \int_0^\pi t\sin^y...

Re: HSC 2018 MX2 Integration Marathon \begin{align*}I&= \int_{-1}^1 \frac{e^{2x}+1 - (x+1)(e^x+e^{-x})}{x(e^x-1)}dx\\ &= \int_{-1}^1 \frac{(e^x-x-1)(e^x+e^{-x})}{x(e^x-1)}dx\\ &= -\int_{1}^{-1}\frac{(e^{-u}+u-1)(e^u+e^{-u})}{-u(e^{-u}-1)}du\\ &= \int_{-1}^1...

I feel like that integral has been on the marathon somewhere before... \text{A bit of experimenting later... and u=sin or u=cos looks problematic} \\ \begin{align*}I&=\int \frac{1}{\sin x \sqrt{\sin 2x}}dx \\&= \int \frac{\sqrt{2\sin x \cos x}}{2 \sin^2x \cos x}dx\\ &= \frac{1}{\sqrt2} \int...

Lol wow that's new. Proving e is transcendental.

Start of prelim "What is circle geometry and why do I need to put up with this cancer" By the end of HSC "FUCK YES CIRCLE GEOMETRY"

Any improvements we could (hopefully) anticipate?

\text{Suppose }u,v\text{ are harmonic and satisfy the Cauchy-Riemann equations in }\mathbb{R}^2.\\ \text{Show that }f=u+iv\text{ satisfies }\\ f^\prime(x) = u_x(x,0) = iu_y(x,0)\text{ for real }x.

That being said, When you first told me that the power series was the definition of the exponential I initially accepted it, but became reluctant to believe. Why is that the case? Or do they just teach things in the wrong order? Because the power series falls out of differentiation as well. I...

At the time I wanted to avoid it because of the fact we hadn't done differentiation but turns out it was the next lecture lol. So I ended up just L'H smashing it. Which is probably me cheating, because this is the definition of the derivative, but I'm in C so I'll cheat by using the...

z\in \mathbb{C} \\\text{I know the answer is 1, but how do I actually compute it? }\\ \lim_{z\to 0}\frac{e^z-1}{z} I can't assume anything about complex differentiability yet

Don't need to show me how to do the entire question if it's way too long. Suggestions are plenty :) f(x)=\int_x^{x+2\pi}\frac{\sin t}{t}dt \text{Prove that }f(x)=O(x^{-2})\text{ as }x\to \infty My starting point was just saying f(x) < integrand being 1/t instead of sint/t, but working...

Oh I see. For our course, "domains" are defined the same way you are, and this was how they defined "regions": A set S is a region if it is an open set together with none, some, or all of its boundary points.