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So I know this is really annoying but I found a really good quote for my paragraph and since I totally suck at English, I need help on identifying what technique is in this quote (if there are any in the first place). The quote: "I’m glad it’s a girl. And I hope she’ll be a fool – that’s the...

Re: HSC 2017 MX2 Integration Marathon $ \noindent IBP yields:\\ \\ $I= \frac{x^2 \sin^{-1}x}{2}-\frac{1}{2}\int \frac{x^2}{\sqrt{1-x^2}}$ d$x \\ \\ =\frac{x^2 \sin^{-1}x}{2}+\frac{1}{2}\int \sqrt{1-x^2}$ d$x-\frac{1}{2}\int \frac{1}{\sqrt{1-x^2}}$ d$x \\ \\ $The first integral can be...

Re: HSC 2017 MX2 Integration Marathon $ \noindent Evaluate $ \int_{0}^{\frac{\pi}{4}} \ln \left ( 1+\tan x \right )$ d$x

Re: HSC 2017 MX2 Integration Marathon $\noindent $I= \int^{\frac{\pi}{2}}_0 \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} $ d$x \\ \\ $ Let $u=\frac{\pi}{2}-x\Rightarrow $ d$u=-$ d$x \\\\ \therefore I=-\int_{\frac{\pi}{2}}^{0}\frac{(\frac{\pi}{2}-u) \sin (\frac{\pi}{2}-u) \cos...

Re: HSC 2017 MX2 Integration Marathon $ let $u^2=1+e^x \\2u$ d$u=e^x $ d$x \\ \\ \therefore I=\int \frac{2u^2}{u^2-1}$ d$u\\ = 2\int $ d$u+\int \frac{2}{u^2-1}$ d$u \\ \\ =2u + \ln \left | u-1 \right |-\ln\left | u+1 \right | \\ \\ = 2\sqrt{1+e^x}+\ln\left | \frac{\sqrt{1+e^x}...

Re: HSC 2017 MX2 Integration Marathon There might be a way to do this by playing around with the numerator, anyways here's how I did it $ Divide each term by $\sin x I=\int \frac{1+\tan x}{3+4 \tan x}$ d$x \\ $ let $u= \tan x \\ $ d$u=\sec^2 x $ d$x \\ \Rightarrow $...

Re: HSC 2017 MX2 Integration Marathon Find: \int \frac{\sin x\cos x}{\sin^4 x+\cos^4 x}$ d$x

Ah thanks for that, didn't know this works for more than three terms

Re: HSC 2017 MX2 Integration Marathon $let $x=\sqrt3 \tan \theta \\ $d$x=\sqrt3 \sec ^2 \theta$ d$\theta \\ \\ I=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{\ln \left |\sqrt3 \tan \theta \right |}{3 +3\tan ^2 \theta}\cdot \sqrt3 \sec ^2 \theta$ d$\theta \\ \\ =...

Well you could form a polynomial with roots alpha^2, beta^2, gamma^2, delta^2. And then finding the sum of roots (-b/a) of that polymomial

A straight forward method to find the real solutions: (2^x-4)^3+(4^x-2)^3=(4^x+2^x-6)^3 \\ $ Let $a\equiv 2^x-4, $ and $b\equiv 4^x-2 \\ \therefore a^3+b^3=(a+b)^3 \\ (a+b)(a^2-ab+b^2)=(a+b)^3 \\ (a+b)(a^2-ab+b^2-a^2-2ab-b^2)=0 \\ -3ab(a+b)=0 \\ \Rightarrow a=0, b=0, a=-b \\ $The first two...

lol I heard from my physics teacher that there was this question in HSC astrophysics to find the distance to a star using the distance-modulus formula. Since alot of people don't know logs and couldn't find the exact distance, the marking criteria was lowered so that it is possible to get full...

Re: HSC 2017 MX2 Integration Marathon $Observe that $\sin x \equiv \sin x + \cos x +(\sin x - \cos x) -\sin x \\ \therefore I = \int \frac{\sin x + \cos x}{\sin x + \cos x}$ d$x+\int \frac{\sin x - \cos x}{\sin x + \cos x}$ d$x -\int \frac{\sin x}{\sin x + \cos x}$ d$x \\ \\ 2I=x-\ln\left |...

Re: HSC 2017 MX2 Integration Marathon My bad, should have wrote it there. I just used reverse quotient rule, with the denominator being 1+lnx and the numerator x Ohhh just realized stupid_girl already done this above

Re: HSC 2017 MX2 Integration Marathon 4. \int \frac{\ln x}{(1+\ln x)^2}$ d$x \\ = \int \frac{\ln x+1-\frac{x}{x}}{(1+\ln x)^2}$ d$x \\ =\frac{x}{1+\ln x}+$ C$

ye

high school

We're currently doing conics too, and its pree fun tbh

Auxiliary! wow thats clever

$let $ \tan \theta=\frac{a}{b}, \tan \phi =\frac{b}{a} \\ \\\tan(\theta + \phi)=\frac{\tan \theta + \tan \phi}{1-\tan \theta \tan \phi} \\ = \frac{\frac{a}{b}+\frac{b}{a}}{1-1}= $undefined/$ \infty \\ \Rightarrow \theta + \phi = \frac{\pi}{2}\\ \therefore...